UC-NRLF 


C   3   250   55E 


js.  NO.  273. 


U.   S.  DEPARTMENT  OP  AGRICULTURE, 
"WEATHER  BUREAU. 


STUDIES  ON  THE  STATICS  AND  KINEMATICS  OF  THE  ATMOSPHERE 

IN  THE  UNITED  STATES. 


Reprints  from  the  Monthly  Weather  Review,  January  to  July,   1002. 


FRANK  H.  BIGELOW,  M.  A.,  L.  H.  D., 

PROFESSOR    OF    METEOROLOGY. 


PREPARED  UNDER  THE  DIRECTION  OF 

WILLIS   L.   MOORE, 

CHIEF  1.   S.   WEATHER  BUREAU. 


WASHINGTON  : 
WEATHER     BUREAU. 

1902. 


W.  B.  No.  273. 


10254 

U.   S.  DEPARTMENT  OF  AGRICULTURE, 

t|,5    "WEATHER  BUREAU.  , 
IU      '  -1 


STUDIES  ON  THE  STATICS  AND  KINEMATICS  OF  THE  ATMOSPHERE 

IN  THE  UNITED  STATES. 


Reprints  from  the  Monthly  Weather  Review,  January  to  July,  1902. 


PRANK  H.  BIGELOW,  M.  A.,  L.  H.  D., 

PROFESSOR    OF    METEOROLOGY. 


PREPARED  UNDER  THE  DIRECTION  OF 

WILLIS    L.    MOORE, 

CHIEF  U.  S.  WEATHER  BUREAU. 


WASHINGTON: 
WEATHER     BUREAU. 

1902. 


Page. 
I.— A  new  barometric  system  for  the  United  States,  Canada,  and 

the  West  Indies  

Preliminary  remarks ... 

The  adopted  standard  elevation  for  the  epoch,  January  1,  1900. 

Other  corrections  to  the  station  pressures 

The  sea  level,  3,500-foot  and  the  10,000-foot  planes  of  reference . 

The  new  reduction  pressure  tables 

Previous  discussions  of  the  plateau  problem 

Bigelow's  system  of  barometry,  1902 

The  sea-level  temperatures 

The  first  pressure  reduction  to  sea  level 4 

To  find  t  —  6 

The  first  process 

The  second  process 5 

The  second  pressure  reduction  to  sea  level 5 

Pressures   computed   on   the    3,500-foot   and   the   10,000-foot 

planes 

The  first  computation  of  .B,  -B2 

The  second  computation  of  Bl  Bt 6 

II. — Method  of  observing  and  discussing  the  motions  of  the  atmos- 
phere   

Introductory  remarks 9 

Notation  and  coordinates 9 

The  axes  of  coordinates 10 

The  azimuth  rotation 10 

The  composition  and  resolution  of  the  vectors  of  motion 11 

The  resolution  of  forces 12 

Vectors  of  motion  in  high  and  low  areas— rectangular  coordi- 
nates    12 

Vectors  of  motion  in  high  and  low  areas — cylindrical  coordinates.  14 

III. — The  observed  circulation  of  the  atmosphere  in  the  high  and 

low  areas 17 

General  description  of  the  vectors  obtained  by  observation ...  17 

Description  of  the  circulation  over  high  and  low  areas 20 

Discussion  of  the  vectors  in  high  areas 20 

Discussion  of  the  vectors  in  low  areas 21 

The  numerical  values  of  the  vectors  23 

IV. — Keview  of  Ferrol's  and  Oberbeck's  theories  of  the  local  and 

general  circulations 27 

General  comparison  of  Ferrel's  and  Oberbeck's  theories 27 

The  supply  of  local  centers  of  heat 27 

Ferrel's  local  cyclone  28 

Ferrel's  solution 29 

The  German  solution  30 

Ferrel's  theory  of  the  general  circulation  over  a  hemisphere   .  31 

Oberbeck's  solution  of  the  general  circulation  32 

V. — Relations  between  the  general  circulation  and  the  cyclones  and 

anticyclone 37 

Unequal  distribution  of  cyclones  in  North  America  and  Europe- 
Asia  37 

Criticism  of  the  canal  theory  of  the  general  circulation 37 

Modification  of  the  canal  theory  39 

The  structure  of  the  anticyclone 41 

Structure  of  the  cyclone 41 

Special  features  of  the  circulation 43 

The  velocities  in  tornadoes 44 

Tiie  waterspout  off  Cottage  City,  Mass.,  August  19,  1896     ....  45 

VI.—  Certain  mathematical  formula-   useful  in  meteorological  dis- 
cussions     47 

The  need  of  a  standard  system  of  formulae  47 

The  general  equations  of  motion 47 

(1)  The  polar  equations  of  motion  on  the  rotating  earth 47 

(2)  The  cylindrical  equations  of  motion  on  the  rotating  earth.  48 
Remarks  on  the  several  terms  in  the  general  equations  of 

motion 48 

Integration  of  the  general  equations  of  motion  in  polar  coordi- 
nates   , , 48 


Expressions  for  the  gradients  of  pressure  

Evaluation  of  the  coefficient      '  and  other  terms  

Evaluation  of  the  gradients  in  polar  coordinates 

The  equation  of  continuity,  and  some  derived  relations 

The  problems  of  the  aqueous  vapor  contents  of  the  atmos- 
phere   

VII. — A  contribution  to  cosmical  meteorology 

General  remarks 

Summary  of  the  discussion  of  1898 

The  magnetic  observations,  1841-1899 

Comparison  of  the  variations  of  the  solar  prominences  with 
those  of  the  terrestrial  horizontal  magnetic  force  for  the 
interval  1874-1900  

The  variations  of  atmospheric  pressure  over  the  entire  earth . 


Table  1. — Direction  and  velocity  of  motion  in  high  and  low  areas — 

rectangular  coordinates  (Tables  34  and  38)  

Table  2. — Total  velocity  in  highs  and  lows  without  regard  to  direc- 
tions (Table  33,  Section  I) 

Table  3. — General  rectangular  components  of  motion  in  high  and 

low  areas  ( Tables  42  and  43) 

Table  4. — Southward  and  eastward  components  of  velocities  in 
highs  and  lows  (Table  33,  Section  II);  mean  normal  com- 
ponents of  velocities  for  the  United  States  (Table  33,  Sec- 
tion III) 

Table  5. — Component  velocities  in  selected  areas  between  high  and 

low  centers  (Table  33,  Section  IV) 

Table  6. — Anticyclonic  and  cyclonic  components  (Tables  44,  46, 

and  45,  47)   

Table  7. — Mean  anticyclonic  and  cyclonic  components  grouped  in 

three  levels  (Table  52) 

Table  8. — Normal  component  velocities  on  six  selected  planes  .... 
Table  9. — Rectangular  and  cylindrical  coordinates  in  high  areas  . . 
Table  10. — Rectangular  and  cylindrical  coordinates  in  low  areas  . . 

Table  11. — Mean  components  on  the  I,  II,  III  circles 

Table  12. — Northward  and  southward  velocities  in  selected  areas. 

Table  13. — Theoretical  west-east  velocities 

Table  14. — Vertical  diminution  of  pressure 

Table  15,    I. — Components  on  the  meridians  due  to  the  rotation  of 

the  earth  

II. — Components  on  the  meridians  due  to  the  relative 

motion  of  the  atmosphere 

Table  16,    I. — First  components  on  the  parallels  due  to  the  rota- 
tion of  the  earth 

II. — Second  components  on  the  parallels  due  to  the  rota- 
tion of  the  earth 

Table  17,    I. — Vertical  components  due  to  the  rotation  of  the  earth. 
II. — Vertical  components  due  to  the  relative  motion  of 

the  atmosphere 

Table  18. — Anticyclonic  and  cyclonic  velocities  at  each  1,000-meter 


Page. 
49 

49 

50 
51 

52 

55 
55 
55 
56 


level 


Table  19. — Application  of  the  formula;  for  a  cyclone 

Table  20. — Dimensions  and  velocities  in  the  waterspout  off  Cottage 
City,  Vineyard  Sound,  Mass.,  August  19,  1896 

Table  21. — Comparison  of  several  determinations  of  the  total  tem- 
perature change  from  the  surface  to  high  levels 

Table  22. — Total  variations  of  the  horizontal  magnetic  force  for 
the  earth  generally,  arranged  in  26.68-day  periods 

Table  23. — The  variations  of  the  annual  mean  atmospheric  pres- 
sures in  many  districts  of  the  earth,  in  units  of  0.001  inch . . 

ILLUSTRATIONS. 

Figure  1. — Comparison  of  the  two  azimuth  systems 

Figure  2. — Example  of  the  graphic  composition  of  wind  vectors.  .  . 

Figure  3. — Plan  of  the  subareas,  azimuths,  and  compass  points, 
adopted  in  high  and  low  areas,  for  the  discussion  of  cloud 
observations 

Figure  4. — Direction  and  velocity  of  motion  in  high  and  low  areas 
of  tho  cirrus  levels  (Chart  15) , , .  , 

iii 


59 
60 


13 
13 


14 


14 
14 
15 

15 
17 
21 
21 
23 
23 
31 
32 

33 
33 
33 

34 
34 

34 

42 
42 

44 

'53 

57 
61 


11 
11 


12 
13 


IV 


Page.  : 
I  '\-j(i  iv  r>.     A  ill  toyolonlo  and  cyclonic  components  of  the  cirrus  level 

(Chart   Hi)  14 

Figure  (i.  —  Adjustcd  mean  vectors  of  direction  and  velocity  of  mo- 
tion in  high  ureas 18 

Figure  7. — Adjusted  mean  vectors  of  direction  and  velocity  of  mo- 
tion in  low  areas 1!) 

Figure  8. — Total  eastward  velocities  in  high  and  low  areas 20 

Figure  9. — Curling  of  the  northward  and  southward  streams  about 

the  centers  of  high  and  low  areas 21 

Figure  10.  — liadiul  and  tangential  components  in  anticyclonic  and 

cyclonic  ureas 24 

Figure  11. — Ferrel's  circulation  in  warm-center  cyclones 2!) 

Figure  12. — Ferrel's  circulation  in  cold-center  cyclones  29 

Figure  13. — Oberbeck's  circulation  in  warm-center  cyclones 30 

Figure  14. — Ferret's  general  cyclone 32 

Figure  15. — Oborbeck's  component  motions  in  the  general  cyclone.  32 

Figure  16. — Ferrel's  component  currents  by  the  canal  theory 37 

Figure  17. — Bigelow's  component  currents  from  the  Weather  Bu- 
reau observations 38 

Figure  18.— Scheme  of  self  regulation  of  the  circulation  by  the  rise 

and  fall  of  the  gradient 40 


Figure  10. — Mixed  system  of  hyperbolic  and  parabolic  components. 

Figure  20. — General  scheme  of  the  structure  of  cyclones 

Figure  21. — Vertical  section  through  the  atmosphere  

Figuro  22. — Illustrating  the  formation  of  the  equation  of  continuity. 

Figure  23. — Total  temperature  fall  from  the  surface  to  high  levels 
by  several  systems 

Figure  24. — Relative  secular  variations  in  the  sun  spots,  the 
European  magnetic  field,  and  the  American  meteorological 
system 

Figure  25. — Variation  of  the  sun-spot  numbers  and  the  amplitude 
area  numbers 

Figure  26. — Semiannual  period  in  the  horizontal  force  of  the  ter- 
restrial magnetic  Held,  arranged  in  six  successive  11-year 
periods  

Figure  27. — Semidiurnal  period  of  the  direct  type 

Figure  28. — Comparison  of  the  solar  prominence  variations  with 
those  of  the  terrestrial  horizontal  magnetic  force  and  the 
atmospheric  pressures  over  the  entirj  earth 

Figure  29. — Positive  and  negative  pressure  variations  over  the 
earth  as  a  whole  for  successive  years,  on  a  s^ale  of  relative 
numbers  . . 


53 


55 
58 


58 
59 


STUDIES  ON  THE  STATICS  AND  KINEMATICS  OF  THE  ATMOSPHERE  IN  THE 

UNITED  STATES.1 


I._A  NEW  BAROMETRIC  SYSTEM  FOR  THE  UNITED  STATES,  CANADA,  AND  THE  WEST  INDIES. 


PRELIMINARY    REMAKES. 

On  January  1,  1902,  at  the  8  a.  m.  observation,  seventy-fifth 
meridian  time,  a  new  system  for  the  reduction  of  the  station 
barometric  pressures  to  the  sea-level  plane,  was  put  in  oper- 
ation for  the  United  States,  Canada,  and  the  West  Indies. 
Ihe  daily  weather  maps  iised  in  forecasting  the  intensity  and 
the  path  of  storms,  and  the  other  allied  phenomena,  are  there- 
fore constructed  upon  a  basis  differing  from  any  hitherto  used. 
Students  who  consult  the  published  weather  maps  should  re- 
member that  the  series  terminating  with  the  above  date  is  not 
comparable  with  the  others  following  it,  the  difference  at  some 
stations  on  the  Rocky  Mountain  Plateau  for  certain  seasons  of 
the  year  amounting  to  several  tenths  of  an  inch  of  pressure 
by  the  mercurial  barometer.  The  problem  of  reducing  the 
pressures  observed  at  stations  located  on  the  Rocky  Mountain 
Plateau  to  sea  level  has  always  been  recognized  as  one  of  un- 
usual scientific  difficulty,  and  it  has  been  under  discussion  in 
the  Washington  Office  at  intervals  ever  since  the  establishment 
of  the  Government  service.  So  far  as  can  be  judged  at  the 
present  writing  the  success  of  the  new  system  is  assured,  and  if 
this  favorable  opinion  is  confirmed  by  continued  use,  it  will 
mark  the  termination  of  thirty  years'  effort  to  solve  this  ques- 
tion in  a  practical  form.  The  other  plateau  districts  of  the 
world,  Mexico,  South  America,  especially  Argentina,  south 
Africa,  Australia,  and  southern  Asia,  will  doubtless  profit  by 
the  experience  of  the  United  States  Weather  Bureau,  on  con- 
sulting the  solution  adopted  for  the  United  States,  Canada, 
and  the  West  Indies. 

Prof.  R.  F.  Stupart,  Director  of  the  Canadian  Meteorologi- 
cal Office,  has  courteously  cooperated  by  supplying  the  neces- 
sary data  for  the  Canadian  stations,  since  the  common  interests 
of  both  countries  require  the  adoption  of  the  same  methods  of 
barometric  reductions.  There  is  no  task  properly  belonging 
to  the  Weather  Bureau  upon  which  more  time  and  labor  has 
been  expended  than  upon  this  problem,  and  the  present  dis- 
cussion is  the  sixth  well  defined  attempt  to  reach  a  satisfactory 
conclusion.  The  importance  of  putting  the  barometric  pres- 
sures on  the  elevated  plateau,  covering  one  third  of  the  terri- 
tory for  which  the  official  forecasts  are  made,  on  a  satisfactory 
scientific  basis,  fully  justifies  this  work,  because  it  is  of  pri- 
mary importance  not  to  attribute  to  weather  conditions  any 
pressure  changes  that  are  in  reality  due  to  the  method  of 
reduction  to  the  plane  of  reference. 

The  eastern  and  central  portions  of  the  United  States  and 
Canada  are  generally  at  levels  less  than  1,000  feet  above  the 
sea,  and  also  the  Pacific  coast  is  at  low  level,  so  that  for  these 
districts  the  barometric  reduction  offers  no  difficulty.  Be- 
tween these,  throughout  the  Rocky  Mountain  region,  there  is 
a  rough  country  where  the  stations  are  at  different  elevations  up 
to  7,000  feet,  where  the  surface  temperature  conditions  range 
enormously,  say  from  —  40°  F.  to  +60°  F.  on  a  single  map  in 

_'  llppi-intcd  from  the  Monthly  Weather  Review  for  January,  11)02. 


extreme  cases,  where  the  prevailing  winds  from  the  Pacific 
Ocean  produce  one  type  of  weather  on  the  western  slopes  of 
the  mountains  and  another  on  the  eastern,  to  say  nothing  of 
the  effect  of  great  arid  districts  between  them,  and  where  the 
configuration  of  the  mountain  valleys,  in  which  many  of  the 
stations  are  located,  relative  to  the  neighboring  ranges  rising 
up  to  12,000  or  14,000  feet  in  some  cases,  causes  various  local 
peculiarities  in  the  behavior  of  the  barometer. 

A  description  of  the  construction  of  our  new  station  pressure 
normals  is  properly  a  preliminary  to  the  solution  of  the  plateau 
problem.  In  the  years  between  1871-1880,  while  the  baro- 
metric network  was  being  extended  over  the  plateau  districts, 
many  of  the  elevated  stations  were  at  the  Army  posts  where  no 
measurement  of  the  altitude  had  been  made,  except  by  the  ba- 
rometer. We  now  know  that  several  of  these  early  elevations 
were  seriously  in  error,  say  from  10  feet  up  to  200  feet,  and  as 
a  change  of  10  feet  in  altitude  corresponds  approximately  to 
0.010  inch  pressure,  the  irregularities  on  the  sea-level  plane 
arising  from  this  source  alone  were  not  inconsiderable.  The 
gradual  extension  of  the  various  surveys  by  the  Government 
over  the  plateau,  together  with  the  railroad  levels  executed 
and  revised  by  the  different  companies,  have  gradually  built 
up  a  system  of  check  levels  at  intersecting  points,  with  accu- 
rate differential  levels  between  them,  so  that  now  the  absolute 
elevations  of  the  several  stations  have  been  determined  with 
much  accuracy.  An  adjustment  of  these  levels  was  made  by 
Prof.  Cleveland  Abbe  in  1871-72;  the  work  was  then  taken  up 
by  the  Geological  Survey,  and  the  latest  results  of  these  surveys 
are  given  in  Gannett's  Dictionary  of  Altitudes,  edition  of  1900. 
The  Weather  Bureau  was  supplied  with  the  corrected  altitudes 
before  the  publication  of  this  report  by  the  Geological  Sur- 
vey, so  that  we  have  had  the  advantage  of  this  data  from  an 
early  stage  in  our  own  work. 

THE    ADOPTED  STANDARD  ELEVATION    FOR  THE   EPOCH,    JANUARY  1,   1900. 

Besides  the  incorrect  actual  elevations  which,  for  one  reason 
or  another,  have  been  adopted  during  thirty  years,  there  have 
been  numerous  changes  in  the  elevation  of  the  local  offices  of 
the  service  at  the  same  station,  involving  many  small  variations 
in  the  altitude  above  sea  level.  A  very  careful  reexamiuation 
of  the  station  records  of  the  respective  stations  showed  that 
it  was  practically  impossible  to  assign  correct  absolute  eleva- 
tions for  the  several  changes  as  referred  to  the  sea  level,  but 
that  it  was  possible  to  discover  the  differences  by  which  the 
successive  changes  in  height  followed  each  other  (that  is,  the 
height  of  the  barometer  in  the  new  office  above  or  below  that 
in  the  old  office),  the  series  of  variations  giving  a  chain  of  steps 
up  and  down  in  the  succession  of  changes.  These  were  care- 
fully determined,  and  they  were  then  applied  to  the  elevation 
occupied  by  the  station  at  the  epoch  January  1,  1900,  so  that 
the  actual  heights  were  thus  found  for  the  respective  inter- 
vals during  which  the  barometer  remained  in  one  position,  and 


they  were  referred  iu  this  way  to  our  latest  and  beat  elevations 
as  given  by  recent  surveys.  Having  adopted  the  elevation  for 
the  station  at  the  given  epoch,  all  the  recorded  actual  pressures 
were  reduced  to  the  elevation  of  1900  by  small  differential 
pressure  corrections,  so  that  the  entire  pressure  system  be- 
comes homogeneous  for  the  station. 

During  the  years  following  1900  a  similar  plan  is  to  be  fol- 
lowed, and  all  pressures  will  be  reduced  back  to  the  standard 
elevation,  so  that  the  series  will  be  maintained  strictly  com- 
parable throughout  the  life  of  the  station  itself.  There  is 
great  advantage  in  this  procedure,  for  two  reasons.  It  was 
found  that  in  the  other  attempts  to  construct  pressure  nor- 
mals the  earlier  computations  were  readjusted  to  the  latest  ele- 
vations at  the  different  dates,  thus  obscuring  the  record  and 
consuming  a  great  amount  of  labor  without  arriving  at  final 
results.  Also,  the  reduction  tables  to  sea  level,  provided  for 
the  use  of  the  stations,  had  to  be  renewed  with  every  removal, 
which  also  consumed  much  time.  On  the  new  plan;  however, 
each  year's  observations  is  added  directly  to  a  homogeneous 
station  system,  and  the  same  reduction  table  serves  without 
modification  in  consequence  of  any  local  changes.  Indeed  it 
is  absolutely  essential  to  reach  such  a  basis  of  operation  in 
meteorology  as  this,  if  there  is  to  be  made  possible  a  scientific 
study  of  the  secular  variations  of  the  weather,  that  is,  the  large 
problem  of  why  and  how  the  seasons,  the  climate,  and  the 
crops,  differ  from  year  to  year,  this  being  the  next  great  prob- 
lem awaiting  practical  meteorology.  Evidently  all  the  cos- 
mical  questions  involving  variations  in  the  radiations  of  the 
sun  must  be  compared  with  as  definite  a  pressure  system  as 
this,  if  scientific  results  are  to  be  secured  from  the  meteoro- 
logical data.  It  may  be  stated  in  passing,  that  in  recent  years, 
since  the  Government  has  erected  large  buildings  in  the  cities 
of  the  United  States,  the  Weather  Bureau  offices  have  been 
more  permanently  located,  and  that  the  average  series  of  un- 
broken observations  is  growing  longer  than  it  used  to  be  10 
and  20  years  ago.  At  the  same  time  the  elevations  are  a  little 
higher,  because  the  offices  are  usually  placed  in  the  upper 
rooms  of  the  lofty  federal  buildings. 

OTHER  CORRECTIONS  TO  THE  STATION  PRESSURES. 

Besides  reducing  the  observed  pressures  to  an  adopted  station 
elevation  it  was  necessary  to  make  several  more  corrections  in 
order  to  obtain  a  homogeneous  system  of  normals.  ( 1 )  The 
records  were  thoroughly  inspected  for  the  several  corrections 
which  ought  to  be  applied  to  the  barometer  readings,  and  we 
have  now  a  complete  list  of  the  barometer  numbers  and  their 
errors  for  capilarity,  scales,  etc.  Besides  eliminating  a  few 
mistakes,  there  were  two  important  special  corrections  to  be 
applied.  During  the  interval  1873-1878  a  correction  of  0.013 
inch  had  been  added  to  the  Signal  Service  standard  barome- 
ter to  reduce  it  to  the  supposed  Kew  standard,  but  a  system 
of  comparisons  instituted  in  1877-78  showed  that  this  was 
probably  an  error,  and  I  have,  therefore,  removed  it  from  the 
new  series.  A  policy  prevailed  in  the  office  from  1888  to  1898 
to  the  effect  that  small  errors  could  properly  be  neglected  in  the 
barometer  reductions,  and  in  accordance  with  it  all  corrections 
for  scale  error  and  capillarity  smaller  tuan  ±  0. 007  inch  were 
discarded;  these  have  now  been  all  restored.  (2)  The  correc- 
tion to  standard  gravity,  at  sea  level  on  the  forty-fifth  parallel 
of  latitude,  was  applied  during  some  years  and  omitted  during 
others,  so  that  there  was  irregularity  in  this  respect.  The 
gravity  correction  has  now  been  systematically  added  by  me 
since  the  beginning  of  1873.  (3)  The  hours  of  simultaneous 
observation  have  been  changed  several  times  since  the  opening 
of  the  service,  but  practically  the  observations  can  be  grouped 
in  two  series  of  selected  hours,  7  a.  m.,  3  p.  m.,  11  p.  m. ,  till 
June  30,  1888,  and  8  a.  m.,  8  p.  m.  since  that  date.  Referred  to 
the  mean  of  24  hourly  observations  which  is  the  natural  stan- 
dard to  adopt  for  the  world,  these  two  systems  present  very 
different  types  of  corrections  for  the  North  American  Conti- 


nent, and  they  must  be  reduced  to  some  one  system  in  order 
to  be  comparable.  Accordingly  auxiliary  tables  were  prepared 
by  which  observations  at  a  few  selected  hours  could  be  re-i 
duced  to  the  mean  of  24  hourly  observations,  and  the  different 
series  have  been  so  corrected  since  January  1,  1873.  These 
24  hourly  corrections  will  be  applied  in  the  future  to  all 
monthly  and  annual  pressures  published  by  the  Weather  Bu- 
reau, so  that  the  fundamental  system  may  remain  intact  in 
case  other  hours  of  observation  should  ever  be  adopted,  differ- 
ing from  those  now  in  use. 

The  application  of  the  corrections  for  local  elevation,  scale 
error,  capillarity,  instrumental  temperature,  gravity,  and  diur- 
nal variation,  to  the  barometric  readings,  gives  a  smooth  homo- 
geneous system  of  values,  from  which  the  mean  annual  and  the 
mean  monthly  station  normals  were  derived  and  checked  by 
cross  addition;  they  are  noted  as  IS.  From  these  the  annual 
and  the  monthly  variations  from  the  general  mean  were  ob- 
tained, and  they  have  been  thoroughly  discussed  in  the  He- 
port  of  the  Chief  of  Weather  Bureau,  1900-1901,  Vol.  II.  In 
order  to  determine  our  final  station  normals,  /!t.  it  was  further 
necessary  to  reduce  all  the  short  series  to  a  standard  fixed  by 
the  long  27-year  series  for  a  large  number  of  stations  sufficient 
to  control  the  work.  There  are  about  two  hundred  and  sixty- 
five  stations,  including  the  Canadians,  to  be  dealt  with,  and  of 
these  about  seventy-five  had  a  long  record  of  twenty-seven  \  ears. 
The  run  of  the  monthly  residuals  increases  in  irregularity  as 
the  number  of  years  of  observation  decreases,  but  \\  e  lia\c  so 
managed  the  discussion  that  a  short  series  normal  can  be  re- 
duced to  the  long  series  normal,  and  thus  the  station  placid 
upon  the  standard  basis.  Whenever  a  new  station  is  opened 
by  the  Weather  Bureau  a  standard  normal  pressure  can  now  be 
constructed  by  a  brief  computation,  and  the  normal  is  more  ac- 
curate than  any  that  could  be  obtained  by  fifteen  years  direct 
observations,  since  these  take  up  all  the  turbulent  pressure 
fluctuations  due  to  the  general  and  local  circulations,  which  it 
is  impossble  to  eliminate,  except  by  the  use  of  the  observations 
of  many  years.  I  may  add  that  my  experience  with  the  baro- 
metric observations  of  the  United  States  convinces  me  that 
they  have  always  been  of  a  high  order  of  scientific  excellence, 
and  that  the  apparent  residuals  are  not  in  fact  due  to  acciden- 
tal irregularities,  but  possess  general  and  even  cosmical  sig- 
nificance when  they  are  thoroughly  discussed.  It  has  been  a 
mistake  to  assume  that  they  are  not  worth  the  most  exact 
treatment  in  the  reductions;  on  the  other  hand  there  is  every 
reason  to  believe  that  they  will  become  of  prime  importance 
in  the  solution  of  several  solar-terrestrial  problems. 

THE    SEA    LEVEL,    3,500-FOOT    AND    THE    10,000-FOOT    PLANES    OF 
REFERENCE. 

Having  obtained  these  reliable  station  pressures  throughout 
the  United  States  and  Canada,  the  plateau  problem  now  conies 
before  us  for  discussion,  in  order  to  reduce  the  pressures  taken 
at  different  elevations  to  the  adopted  planes  of  reference,  in 
our  case  to  the  sea-level  plane,  to  the  3,500-foot  plane  and  the 
10,000-foot  plane.  All  the  forecasting  problems  have  been 
heretofore  studied  solely  on  the  sea-level  plane.  But  it  seems 
evident  that  our  grasp  upon  the  weather  problem  will  bo 
greatly  strengthened  if  we  can  study  at  least  three  sections 
through  the  atmosphere  daily  instead  of  the  one  at  the  bottom 
of  it.  I  selected  the  3,500-foot  plane  because  this  is  the  aver- 
age height  of  the  Rocky  Mountain  stations,  to  which  the  least 
possible  reduction  is  required;  also,  because  it  is  the  average 
altitude  of  the  base  of  the  cumulus  cloud  sheet  over  the  eastern 
districts,  upon  which  observations  can  be  most  favorably  made 
with  theodolites  for  gradients  of  pressure,  temperature,  and 
vapor  tension.  Besides  this,  it  is  the  altitude  at  which  the 
moving  currents  of  air  are  sufficiently  distant  from  the  ground 
to  take  on  their  natural  configuration  when  freed  from  the 
surface  turbulent  friction.  The  10,000-foot  plane  was  chosen 
because  it  is  already  in  use  by  the  MONTHLY  WEATHER  REVIEW 


to  show  the  monthly  mean  isobars  at  a  considerable  altitude. 
Furthermore,  it  is  just  in  the  midst  of  the  most  rapid-moving 
horizontal  local  currents,  which  build  up  the  cyclones  and  anti- 
cyclones of  the  middle  latitudes  and  upon  which  the  intensity 
of  storms  depends.  We  know  that  the  isobars  on  these  three 
planes  differ  considerably  from  one  another,  the  closed  curves 
on  the  lower  plane  tending  to  open  out  into  long  sweeps  on 
the  upper  plane,  and  it  is  probable  that  an  interconiparison  of 
these  varying  isobars  from  map  to  map  will  be  valuable. 

THE    NEW    REDUCTION    PRESSURE    TABLES. 

It  is  evidently  necessary  to  possess  reduction  tables  of  a 
perfectly  general  and  flexible  kind  in  order  to  make  the  neces- 
sary reductions  from  the  several  stations  to  these  three  planes, 
and  from  one  plane  to  the  other,  in  either  direction  upward 
or  downward.  As  there  are  no  such  tables  in  print,  I  have 
first  computed  logarithmic  reduction  tables  in  English  meas- 
ures, similar  to  those  in  metric  measures,  described  in  the 
International  Cloud  Report  of  1898-99,  the  intervals  being  for 
every  100  feet  up  to  10,000,  and  for  every  10°  F.  from  —40° 
to  +  100°.  From  these  general  tables  the  special  station 
tables  were  made,  giving  the  corrections  to  be  applied  to  the 
station  pressure  at  intervals  of  0.20  inch  to  reduce  it  to  the 
three  planes,  respectively.  These  individual  tables  contain  a 
correction  for  the  humidity  term  separated  from  the  dry-air 
term,  a  correction  for  the  plateau  effect,  a  residual  reduction 
for  a  few  stations,  and  two  temperature  arguments — rirst,  the 
mean  temperature  of  the  air  column,  and  second,  the  cor- 
responding surface  temperature,  which  is  the  mean  value  of 
two  successive  8  o'clock  observations,  the  last  always  including 
that  hour  for  which  the  reduction  is  made.  In  order  to  sim- 
plify matters  as  much  as  possible  for  the  observers  on  the 
stations,  the  individual  station  tables  are  constructed  by  com- 
bining all  these  corrections  and  applying  them  at  short  inter- 
vals of  the  station  pressure,  namely,  for  every  tenth  of  an 
inch,  and  for  such  close  intervals  of  the  temperature  argument 
that  there  shall  be  no  interpolation  necessary  in  this  direction 
in  order  to  obtain  the  hundredth  of  an  inch  of  reduced  pressure. 
The  result  is  contained  in  three  tables,  one  for  each  plane,  with 
the  surface  temperature  and  station  pressure  as  the  arguments, 
and  the  reduced  pressure  to  the  three  planes,  respectively,  in 
the  body  of  the  table,  instead  of  the  correction  to  the  observed 
pressure.  There  is  thus  no  computation  to  be  done  at  the 
station  to  reduce  the  observation,  and  the  time  consumed  in 
preparing  this  portion  of  the  cipher  code  message  is  very 
short.  The  special  tables  for  the  stations  for  use  in  reduction 
to  the  3,500-foot  and  the  10,000-foot  planes  are  now  being 
made  up,  the  first  and  the  second  forms  leading  up  to  them 
being  completed  and  checked.  The  tables  for  reduction  to 
sea  level  are  already  in  operation,  and,  so  far  as  known,  there 
is  no  occasion  to  modify  the  reductions  at  any  of  the  stations. 
When  one  considers  the  large  amount  of  painstaking  and 
careful  labor  required  to  produce  such  a  result  as  this  in  so 
complex  a  problem,  it  is  a  pleasure  to  commend  the  faithful 
work  of  Mr.  Heiskell  and  Miss  Hawkins,  who  have  been  my 
assistants  in  this  computation.  We  hope  to  be  able  to  make 
a  trial  of  the  working  of  the  pressures  on  the  higher  planes 
before  very  long. 

PREVIOUS    DISCUSSIONS    OF    THE    PLATEAU    PROBLEM. 

After  all  these  preliminary  matters  have  been  concluded  we 
may  proceed  to  the  really  difficult  portions  of  the  work.  They 
group  themselves  around  three  points,  (1)  the  proper  relation 
between  the  observed  surface  temperature,  t,  and  the  mean  tem- 
perature of  the  air  column,  It,  corresponding  to  and  substituted 
for  the  plateau  at  the  regular  intervals  for  which  the  general 
logarithmic  reductions  were  computed;  (2)  the  effect  of  the 
plateau  itself  upon  the  free  air  pressure;  (3)  the  residual  local 
effects  which  can  not  be  classified  with  the  other  reductions. 
Those  will  become  clearer  to  the  reader  by  briefly  mentioning 


the  previous  methods  which  have  been  employed  in  reducing 
the  plateau  pressures  to  the  sea-level  plane.  (1)  From 
1871  to  June,  1881,  the  old  Guyot  tables  were  used  in  reducing 
low-level  stations,  with  the  surface  temperature  and  pressure 
at  the  time  of  observation  as  the  argument.  Certain  annual 
constants  wore  employed  in  the  cases  of  high  stations.  The 
effect  was  to  cause  the  isobars  to  swing  widely  between  the 
morning  and  evening  hours,  and  generally  the  maps  were  very 
unsteady.  (2)  July,  1881,  to  June,  188(5,  monthly  constants  were 
used  for  each  station,  as  recommended  by  the  first  board  on 
barometer  reductions;  a  single  constant  answered  for  each 
mouth;  these  are  sometimes  known  as  the  Abbe-Upton  con- 
stants. (3)  July,  1886,  to  June,  1887,  the  entirely  new  system  of 
tables  by  Professor  Ferrel  was  used,  thus  introducing  several 
valuable  principles.  Thus,  the  mean  temperature  of  the  preced- 
ing twenty-four  hours  was  used  instead  of  that  belonging  to  the 
respective  hours  of  observation;  this  was  reduced  by  a  vertical 
temperature  gradient,  0.165  per  100  feet,  to  the  approximate 
mean  of  the  column ;  the  pressure  and  temperature  arguments 
(B,  t)  were  both  employed  in  entering  the  table ;  a  special  cor- 
rection for  the  plateau  effect  was  made  in  the  form  C,  J  0,  H, 
where  C'=  0.00105,  J  0  is  the  variation  of  the  temperature  from 
the  annual  mean,  and  H  is  the  altitude  in  units  of  a  thousand 
feet.  The  application  of  the  correction  for  the  plateau  effect 
removes  the  wide  range  in  pressure  which  occurs  on  the  plateau 
between  summer  and  winter  and  reduces  it  to  about  the  same 
value  on  the  plateau  and  in  the  low  level  eastern  districts. 
For  example,  if  the  mean  annual  temperature  is  50°,  that  for 
January  25°,  and  for  July  80°,  at  a  station  5,000  feet  above 
the  sea" level,  we  have  0.00105  x  (  —  25)  x  5  =  —0.131  inch  for 
January,  and  0.00105  x(  +  30)x  5=  +0.158  inch  for  July. 
The  annual  range  for  high  stations  on  the  plateau  is  about 
0.400  inch,  and  on  the  low  levels  it  is  only  0.150,  the  difference 
being  simply  the  plateau  effect.  Professor  Ferrel's  tables  were 
not  used  very  long.  (4)  July,  1887,  to  December,  1890,  a  mix- 
ture of  Ferrel's  and  Hazen's  tables;  1891-1901,  Hazen's  tables. 
Professor  Hazen  constructed  a  general  empirical  formula  with 
the  object  of  simplifying  the  form  of  the  station  table.  For 
this  purpose  he  assumed  that  Mount  Washington  is  the  type 
for  the  plateau  reductions,  which  is  in  fact  erroneous,  since 
that  isolated  mountain  acts  like  a  free  air  point,  except  for  the 
modified  value  of  0;  he  assumed  that  the  sea-level  pressure 
should  always  be  exactly  30.00  inches,  and  at  the  same  time 
abandoned  the  pressure  argument  entirely,  with  all  depending 
upon  it,  and  computed  the  correction  under  these  conditions; 
he  rejected  the  plateaii  effect  correction,  and  at  the  same  time 
the  change  of  surface  temperature  to  the  mean  temperature, 
0,  was  neglected.  On  applying  this  system  to  the  daily  map 
it  was  necessary  to  make  certain  arbitrary  changes  in  the  com- 
puted reductions  in  order  to  produce  smooth  isobars.  The 
great  simplicity  in  the  use  of  this  table,  having  only  the  sur- 
face temperature  as  argument,  seems  to  have  been  considered 
sufficient  ground  for  substituting  these  tables  for  Ferrel's,  so 
that  from  1891  till  1901,  inclusive,  they  have  been  employed 
in  making  the  daily  weather  maps,  although  well  known  to  be 
unscientific  and  inaccurate.  However,  it  should  be  said  that 
although  the  plateau  correction  was  omitted,  the  practical 
working  of  the  Hazen  method  was  such  as  to  make  the  sea- 
level  reductions  conform  much  more  closely  to  the  Ferrel  sys- 
tem than  to  the  pure  Laplacean  system,  which  is  correct  for 
free  air  rediictions  only.  On  this  account  the  Ferrel  and  the 
Hazeu  systems  work  in  the  same  direction  for  wide  departures 
of  the  temperature  from  the  annual  mean,  and  to  some  extent 
relieve  the  plateau  exaggeration,  so  that  we  conclude  that  the 
weather  maps  have  served  fairly  well  for  the  practical  purposes 
of  forecasting.  (5)  1895-1896,  Professor  Morrill,  in  connec- 
tion with  a  second  board  on  barornetry,  rediscussed  the  prob- 
lem and  computed  a  set  of  tables  which  have  not  been  pub- 
lished, though  they  have  been  used  for  some  office  work,  espe- 
cially the  construction  of  the  sea  level  and  the  10,000-foot 


plane  maps  for  the  MONTHLY  WKATHER  HKVIKW  (luring  1896-1901. 
The  Laplacean  free  air  reduction  was  computed  by  special 
tables  for  the  pressure  and  the  temperature  arguments,  the 
value  of  0  being  found  by  certain  adopted  average  vertical 
gradients  varying  for  the  different  seasons  of  the  year;  the 
humidity  term  was  made  so  as  to  modify  the  logarithmic  argu- 
ment; the  plateau  term  was  entirely  omitted;  the  tables  were 
in  the  form  of  a  logarithmic  argument,  which  was  not  very 
convenient  for  rapid  work.  It  was  suggested  at  the  same  time 
that  a  system  of  constants,  daily  rather  than  monthly,  be  re- 
sumed for  making  the  necessary  forecasting  isobars. 

BIGELOW'S  SYSTEM  OF  BAROMETRY,   1902. 

We  now  come  to  the  sixth  attack  upon  the  problem,  and 
shall  here  merely  enumerate  the  steps  in  the  discussion,  while 
the  report  itself  will  be  found  in  Volume  II,  Annual  Report, 
Chief  of  Weather  Bureau,  1900-1901.  In  substance  the  prin- 
ciples laid  down  by  Ferrel  have  been  adopted,  but  the  work 
has  been  carried  far  beyond  the  degree  of  perfection  possible 
to  him  nearly  twenty  years  ago,  in  consequence  of  the  numer- 
ous observations  at  our  disposal,  whereas  Professor  Ferrel 
contented  himself  with  only  four  years  of  observation  at  the 
plateau  stations  preceding  the  time  of  his  studies. 

THE   SEA-LEVEL    TEMPERATURES. 

The  object  to  be  obtained  is  to  separate  the  temperature 
argument  from  the  plateau  effect,  and  to  arrive  at  smooth 
isobars  in  correct  relations  to  the  winds  and  the  weather 
throughout  the  Rocky  Mountain  region.  Having  prepared 
the  monthly  station  pressure  normals,  as  described  above,  the 
corresponding  station  temperature  and  vapor  tension  normals 
were  extracted  from  the  office  records.  The  plateau  is  there- 
fore to  be  considered  as  dotted  over  with  GO  or  70  stations 
where  the  monthly  values  of  the  elements  (It,  t,  e)  are  known. 
Assuming  an  average  vertical  temperature  gradient  of  0.30°  per 
100  feet,  the  temperatures  were  first  reduced  from  those  given 
at  the  station  elevation,  to  corresponding  values  at  selected 

heights,  500,  1,500, 6,500  feet,  through  short  distances; 

for  example,  all  between  0  and  1,000  feet  were  corrected  to 
500  feet,  and  so  on.  This  concentrates  the  reductions  on  a 
few  planes.  Then  a  preliminary  set  of  temperature  gradients 
in  latitude  and  longitude  was  computed  from  the  temperatures 
on  these  few  planes,  throughout  the  region  west  of  the  Mis- 
sissippi Valley.  Certain  centers  of  reduction  were  taken,  namely, 
where  the  one  hundred  and  twentieth  meridian  crosses  the 
fiftieth,  forty-fifth,  fortieth,  and  thirty-fifth  parallels  of  lati- 
tude, and  the  one  hundred  and  tenth,  one  hundredth,  and 
ninetieth  meridians  cross  the  same  parallels,  and  the  tempera- 
tures were  reduced  by  the  two  horizontal  gradients  to  these 
centers,  so  that  a  series  of  temperatures  varying  with  the  alti- 
tude are  now  known  in  vertical  directions,  at  about  18  geo- 
graphical points.  These  temperatures  were  plotted  on  a 
diagram  whose  abscissas  are  temperature  values  and  whose 
ordinates  are  altitudes,  one  chart  for  each  month  and  one  for 
the  year;  average  curves  were  drawn  through  the  plotted  tem- 
peratures and  prolonged  by  best  judgment  to  the  sea  level. 
In  the  majority  of  cases  it  was  easy  to  do  this,  as  the  curvature 
was  distinctly  developed  on  the  diagrams.  In  this  way  sea-level 
temperatures  were  found  at  several  evenly  distributed  points 
beneath  the  plateau,  and  they  were  transferred  to  monthly  charts 
on  the  sea-level  plane,  which  were  completed  for  the  Pacific  low 
level  districts  and  for  the  central  and  eastern  portions  of  the 
United  States  and  Canada.  A  system  of  well  graded  isotherms 
was  drawn  through  them  for  the  entire  country.  Small  ad- 
justments of  the  temperatures  on  the  centers  of  reduction 
were  required  to  make  the  temperatures  of  the  vertical  system 
and  of  the  horizontal  system  interlock  harmoniously  and  agree 
together  on  the  sea-level  plane.  Furthermore,  new  and  more 
accurate  temperature  gradients  in  latitude  and  longitude 
could  now  be  obtained,  and  the  work  was  therefore  repeated 


from  the  beginning  to  the  end  with  the  improved  values.  The 
adopted  temperature  system  is  the  result  of  two  or  three  such 
approximating  computations,  so  that  it  has  at  last  sutlicient 
reliability  to  become  the  substantial  basis  for  further  reduc- 
tions. The  sea-level  temperatures  at  the  se\eral  stations  can 
easily  be  scaled  from  these  charts  to  the  tenth  of  a  degree, 
and  such  values  are  called  ta.  The  use  of  centers  of  reduc- 
tion commends  itself  by  the  fact  that  the  stations  can  l>e 
grouped  in  several  ways,  since  the  same  station  can  be 
reduced  to  different  centers,  and  the  local  inaccuracies  will 
thus  check  themselves  out;  also  by  the  fact  that  the  entire 
amount  of  computation  is  much  smaller  and  its  accuracv  can 
be  controlled  by  the  algebraic  differences  for  uniform  spaces. 
The  most  important  result  of  this  discussion  is  the  develop- 
ment of  well  defined  temperature  inversions  during  the  winter 
on  the  northern  Rocky  Mountain  slope,  and  in  the  summer  in 
the  southern  California  districts.  The  former  are  dm  to  the 
dynamic  heating  of  the  air  blowing  eastward  <>\er  the  liockv 
Mountain  divide,  and  the  latter  to  the  excessive  surface  heat- 
ing of  the  arid  region  relatively  to  the  temperature  of  the  Pa- 
cific Ocean.  The  introduction  of  these  inversion  gradients 
relieves  the  congestion  of  the  isothermal  lines  heretofore  drawn 
in  these  districts. 

THE    FIRST    PRESSURE    REDUCTION    TO    SEA    LEVKI,. 

Finally,  the  relative  humidities  were  assumed  to  be  the 
same  for  the  surface  and  the  sea-level  plane  throughout  the 
plateau,  and  from  the  values  of  ta  just  found  the  corresponding 
sea-level  vapor  tensions,  «o,  were  computed.  We  have  thus 
obtained  all  the  elements  required  for  a  reduction  of  the  sur- 
face pressure,  B,  to  the  sea-level  pressure,  />' ,  by  taking  as 

t  -4-  t  i' 

a  first    approximation  0  =  —  ~?  and  the  ratio     ;,  where   li    is 

2  j> 

nearly  30.00  inches  for  all  monthly  means.  Using  our  new 
logarithmic  tables,  the  monthly  and  annual  pressure  at  each 
station  in  the  United  States  and  Canada  was  reduced  to  sea 
level,  and  the  results  were  transferred  to  charts.  Isobars 
were  drawn  through  these  sea-level  pressures  as  accurately  as 
the  data  permitted,  though  the  values  of  7.  were  quite  dis- 
cordant in  many  places,  and  the  lines  somewhat  in  doubt.  Of 
course  the  plateau  correction  was  included  in  the  sea-level  re- 
duction, as  stated  above. 

TO  FIND  t  — 0. 

For  practical  working  by  the  tables,  it  was  first  necessary  to 
determine  the  relations  of  t  and  0  for  the  entire  range  of  tem- 
peratures throughout  the  year,  and  this  was  a  task  of  no  little 
perplexity.  It  was,  however,  finally  accomplished  by  two 
processes.  It  will  be  remembered  that  in  the  Abbe-Upton 
system  of  monthly  constants  and  in  the  Hazeu  empirical  tables 
this  modification  of  the  surface  temperature  argument  was 
omitted;  that  Ferrel  used  a  constant  vertical  gradient  of 
0.165°  per  100  feet  for  the  year  to  pass  from  t  to  <>,  and  that 
Morrill  modified  this  gradient  by  taking  per  100  feet,  0.150° 
in  winter,  0.200°  in  spring  and  autumn,  and  0.250°  in  summer. 
My  vertical  temperature  gradient  came  out  about  0.195°  for 
each  month  in  the  year,  as  the  average  for  the  entire  plateau, 
but  it  was  distinctly  shown  that  the  several  portions  of  the 
plateau  have  very  different  gradients  in  the  same  mouth,  and 
that  for  the  same  locality  they  change  greatly  from  month  to 
month.  Hence  it  was  improper  to  attempt  to  deal  with  the 
plateau  as  a  whole  by  using  the  same  temperature  gradient; 
so  that,  in  fact,  each  station  must  be  considered  not  only  by 
itself,  but  also  in  its  relations  to  the  neighboring  stations. 
Finally,  special  curves  have  been  constructed  for  temperatures 
between  —  40°  F.  and  +  100°  F.,  showing  the  variable  differ- 
ence between  t,  the  surface  temperature  for  twenty-four  hours, 
and  the  corresponding  0,  or  the  mean  air  temperature  of  an 
air  column  substituted  for  the  plateau  itself.  The  0  can  not 


be  considered  as  the  arithmetical  mean  temperature  between 
the  surface  and  the  sea-level  temperatures,  because  the  con- 
necting line  is  a  curve  and  is  not  straight,  so  that  it  is  essen- 
tial to  arrive  at  an  integral  mean  temperature  instead  of  an 
arithmetical  mean.  In  a  graphical  construction  the  values  of  0 
may  be  taken  as  the  abscissae  and  the  differences,  t  —  0,  as  the 
ordiuates  of  a  curve,  which  we  seek  to  construct.  The  first  ap- 
proximation is  evidently  equal  tot  —  0  =  t  —  \(t-\-  <„)  =  J  (t  —  t0), 
but  the  true  value  may  differ  from  this  by  several  degrees  at 
many  of  the  high  stations. 

THE    FIRST    PROCESS. 

We  proceeded  to  discuss  this  point  by  two  distinct  methods, 
the  first  covering  the  low  temperatures  from  — 40°  to  -(-30°, 
and  the  second  covering  the  temperatures  from  about  10°  to 
90°,  so  that  there  shall  occur  a  small  overlapping  of  the  two 
systems  in  the  middle  temperatures,  and  thus  allow  the  two 
to  be  joined  together.  About  fifty  maps  were  selected  for  the 
winter  season,  when  high  pressures  and  low  temperatures  pre- 
vailed in  the  Rocky  Mountain  districts.  The  pressures  for  the 
plateau  stations  were  next  reduced  to  the  3,500-foot  plane,  be- 
cause this  requires  the  least  average  run  for  the  corrections, 
and  hence  there  is  little  error  arising  from  selecting  the  wrong 
temperature  arguments.  This  configuration  of  isobars  was 
drawn  in  red  lines;  then  the  low  stations  near  the  Pacific 
Ocean  and  those  in  the  Mississippi  Valley  were  reduced  to  sea 
level;  also  some  of  the  stations  on  the  mountain  slope  at  mod- 
erate elevations  were  reduced  to  the  3,500-foot  plane  as  well 
as  to  sea  level.  A  set  of  isobars  was  drawn  on  the  sea  level 
in  blue  lines.  It  was  now  assumed  that  the  configuration  on 
the  3,500-foot  plane  is  substantially  correct  for  that  elevation, 
and  is  what  the  forecaster  really  wants  at  sea  level  for  practical 
work.  It  was  therefore  joined  with  the  sea-level  system  by 
simply  making  the  red  and  blue  lines  flow  together  and  uniting 
them  smoothly;  in  other  words,  the  upper  configuration  was 
depressed  to  sea  level  by  simply  renumbering  the  isobars  in 
inches  as  determined  by  the  true  sea-level  lines,  so  that  a 
single  system  of  well-balanced,  isobars  covered  the  country. 
Next  the  question  was,  what  is  the  value  of  0  that  will  be  re- 
quired to  transform  the  observed  station  pressure  into  the 
sea-level  pressure  thus  constructed  ?  This  was  computed  from 
the  data  in  a  reverse  direction,  and  the  differences,  t  —  0,  found ; 
these  were  collected  by  groups  for  each  station  on  the  plateau 
above  1,000  feet  in  elevation;  the  means  were  taken  and  plotted 
as  ordinates  on  the  abscissa  axis  of  0.  The  result  was  very 
instructive,  and  it  at  once  separated  the  plateau  into  groups 
corresponding  to  the  geographical  and  climatic  location,  and 
showed  that  all  the  attempts  to  use  one  value  of  the  vertical 
gradient  for  a  given  time  are  very  erroneous.  It  should  be  re- 
marked that  the  value  of  t  —  0  thus  found  was  much  too  large, 
because  it  included  within  itself  the  real  plateau  effect,  and 
this  ought  first  to  have  been  separated ;  but  it  gave  true  rela- 
tive variations  of  t  —  0  with  the  range  of  temperature  from 
— 40°  to  +30°,  so  that  it  was  only  necessary  to  discover  the 
reduction  factor  to  make  the  scale  of  values  correct. 

THE    SECOND    PROCESS. 

For  the  warmer  temperatures  of  the  year,  from  +  30°  to 
-)-  1)0°,  I  took  the  mean  monthly  values  of  t  and  ta,  surface  and 
sea-level  temperature,  respectively,  and  found  t  —  0  =  \  (t.  —  ta) 
and  (t  =  |  (t  +  ta).  These  were  plotted  mouth  by  month  in 
coordinate  points  through  which  it  was  easy  to  draw  approxi- 
mate mean  curves.  It  is  noted  that  during  the  winter  months 
the  ordiuates  average  a  little  larger  for  the  same  values  of  0 
than  during  the  summer  months,  but  as  we  are  limited  to  con- 
structing a  set  of  tables  representing  mean  conditions,  this 
mean  line  is  the  best  that  can  be  taken.  The  variation  on  the 
mean  line  does  not  often  exceed  ±  1°,  and  this  small  change  in 
the  resulting  argument  has  really  but  little  influence  upon  the 
sea-level  reductions  which  are  required.  Finally  the  slope  of 


the  second  system  of  curves  at  the  temperatures  from  +  10  to 
-f-  30°  indicated  the  slope  that  should  be  assigned  to  those  found 
by  the  first  method,  that  is,  they  gave  us  the  scale  factor  for 
reducing  the  slope  first  obtained.  The  resulting  curves  are 
published  in  the  full  report,  but  they  can  hardly  be  described 
without  diagrams.  Generally  speaking,  on  the  north  and  east 
of  the  plateau  the  (t — 0)  curves  have  a  short  ordinate  from 
10°  to  40°,  and  a  considerable  increase  toward  either  end; 
on  the  central  portions  of  the  plateau  the  curves  are  nearly 
flat,  the  length  of  the  ordinates  being  about  proportional  to 
the  altitude;  on  the  western  side  of  the  plateau  the  curves 
have  ordinates  which  are  longest  in  the  central  parts  and  short- 
est at  the  ends,  that  is  to  say,  they  are  about  reversed  in  shape 
from  those  on  the  eastern  plateau.  These  differing  results  are 
largely  due  to  the  climatic  effects  of  prevailing  winds  from  the 
Pacific  Ocean,  which  blow  upon  the  mountain  ranges  and  pre- 
cipitate their  moisture  on  the  western  side;  the  clear  skies  and 
cold  waves  prevail  on  the  eastern  side;  also  there  are  seen  to 
be  certain  dynamic  heating  effects.  This  subject  is,  however, 
too  large  to  expand  in  this  connection. 

THE   SECOND    PRESSURE    REDUCTION    TO    SEA    LEVEL. 

Equipped  with  these  first  approximate  values  of  0  for  each 
month  as  derived  from  the  surface  t,  the  reductions  to  sea  level 
were  made  for  the  mean  monthly  normal  station  pressures,  Ba, 
as  already  mentioned,  and  the  corresponding  isobars  were 
drawn.  The  sea-level  pressures,  as  shown  by  the  resulting 
map  itself,  apart  from  the  reduced  values,  are  really  more 
nearly  well  balanced  and  correct  than  those  derived  from  the 
individual  reductions,  because  the  isobars  depend  upon  the 
mean  result  of  many  neighboring  stations,  whose  mutual  claims 
must  be  simultaneously  satisfied  in  drawing  the  pressure  lines. 
The  pressures  were,  therefore,  scaled  from  the  maps,  giving  Bm, 
and  the  differences  taken  between  them  and  the  original  values 
as  reduced  by  the  computation  for  Ba-Bm.  The  outcome  was  ex- 
ceedingly valuable  and  suggestive.  For  some  stations  the 
differences  between  the  map  and  reduced  values  were  such  as 
to  indicate  only  minor  irregularities  of  a  few  thousandths 
of  an  inch,  and  these  are  to  be  referred  to  imperfections  in  the 
station  normals;  for  others  the  difference  was  nearly  constant, 
suggesting  an  error  in  the  assumed  elevation,  especially  for  the 
old  stations  at  military  posts  where  the  elevation  had  been 
derived  from  barometer  readings;  for  others  there  was  a  very 
marked  annual  period  in  the  differences,  which  could  only  be 
due  to  an  error  in  the  assigned  value  of  the  mean  temperature, 
t>,  since  the  differences  disappeared  at  certain  points,  the  signs 
being  reversed  between  the  low  and  the  high  temperatures. 
To  be  brief,  all  these  sources  of  difference  were  removed,  the 
entire  work  was  recomputed  a  second  time,  a  new  system  of 
isobars  was  drawn,  and  generally  the  entire  subject  was  worked 
over  in  every  available  way.  The  practical  effect  was  a  readjust- 
ment of  some  elevations,  and  of  the  values  of  0,  so  that  the 
final  differences  between  the  map  and  the  reduced  sea-level 
pressures  became  small,  usually  less  than  one  hundredth  of  an 
inch  (0.010  inch),  for  the  long  record  stations.  In  a  few  cases  it 
was  found  that  the  constant  error,  called  A  A,  was  due  to  the 
fact  that  the  initial  temperature  from  which  the  plateau  cor- 
rection was  computed,  namely,  C  A  0  H,  was  not  accurately 
chosen.  Usually  this  was  taken  as  a  mean  annual  tempera- 
ture, but  for  some  stations,  especially  on  the  southwestern  edge 
of  the  plateau,  Santa  Fe,  Flagstaff,  Modena,  Independence, 
etc.,  it  should  have  been  somewhat  different.  The  variation 
can  not  be  due  to  elevation,  because  this  has  been  carefully 
determined  by  the  surveys,  but  it  must  be  caused  by  the  local 
influence  of  the  great  desert  in  connection  with  the  adjacent 
lofty  mountain  ranges.  There  are  other  stations  of  low  eleva- 
tion, lying  in  the  eastern  or  in  the  Pacific  coast  districts,  where 
no  important  error  can  arise  from  the  reduction  data,  at  which 
there  is  a  small  constant  correction  required  to  make  the  station 
harmonize  with  the  others,  as,  for  example,  Lynchburg.Va.,  and 


Portland,  Me.  These  stations  have  been  known,  at  the  Central 
Office,  to  act  out  of  perfect  harmony  with  their  surroundings. 
and  it  is  still  difficult  to  understand  the  causes  of  these  discrep- 
ancies. It  has  been  found,  furthermore,  that  the  low  station:- 
on  the  north  Atlantic  and  south  New  England  coast  and  also 
on  the  north  Pacific  coast,  are  not  so  perfectly  in  accord  as 
might  be  expected,  and  this  may  be  due  to  the  effect  of  some 
land  and  sea  action  which  is  operating  in  these  localities.  On 
the  whole  the  reductions  as  completed  are  very  reliable  when 
all  corrections  are  applied,  that  is  to  about  0. 010  inch,  under 
all  possible  circumstances.  We  note  further  that  the  differ- 
ences outstanding  between  the  finally  adjusted  reductions  to 
sea  level  from  the  station  normal  pressures  and  the  map  pres- 
sures derived  from  the  balanced  system  of  isobars,  can  be 
properly  considered  as  corrections  to  the  station  normals 
which  will  reduce  them  to  the  homogeneous  or  balanced  nor- 
mals. This  is  distinctly  true  for  stations  of  short  record,  e.  g. , 
two  or  three  years,  where  the  monthly  variations  are  really 
considerable,  so  that  by  applying  these  residuals  as  corrections 
the  station  normals  are  brought  to  agree  with  the  more  correct 
system  which  would  be  derived  from  a  long  record  of  obser- 
vations. In  short,  since  the  long  record  stations  really  control 
the  map  construction,  the  short  records  can  be  at  once  im- 
proved by  applying  these  small  final  residuals.  Such  residual 
corrections  have,  therefore,  been  added  to  all  station  normals, 
and  the  entire  system  is  thus  reduced  to  a  long  range  homo- 
geneous system  and  it  is  called  Bu,  normal  pressure  at  the  sta- 
tion, and  Bm,  normal  pressure  at  the  sea  level.  These  values 
become  our  standard  normals  for  further  developements  and 
have  been  so  used  in  the  remainder  of  the  work.  It  is  also 
evident  that  whenever  a  new  station  is  opened,  we  can  easily 
compute  a  more  correct  station  normal  pressure,  by  starting 
with  the  values  of  7?m  as  interpolated  from  the  map,  than  could 
be  found  by  less  than  fifteen  or  twenty  years  of  observations. 

PRESSURES    COMPUTED    ON    THE    3, 500-FOOT    AND    THE    10, 000-FOOT 

PLANES. 

We  have  now  obtained  the  following  quantities:  At  the  sta- 
tions, Bf,  t,  e,  R.  H.,  normal  pressure,  temperature,  vapor  ten- 
sion, and  relative  humidity;  on  the  sea-level  plane,  B  ,  t ,  ea; 


also  the  ratio  —  was  computed  for  use  in  the   reductions. 

£> 


It 


is  next  proposed  to  compute  -B,>  <,,  ev  on  the  3,500-foot  plane, 
and  #„  <2,  er  on  the  10,000-foot  plane.  For  this  purpose  the 
temperature  gradients  in  the  free  air  must  first  be  determined. 
There  are  three  sources  of  information  available,  namely,  the 
European  balloon  ascensions,  the  American  kite  ascensions, 
and  the  Washington  gradients  derived  from  computation  on 
the  cloud  formations  observed  with  the  theodolites  in  1896-97. 
These  were  all  thoroughly  discussed  and  they  agree  together 
sufficiently  well  to  permit  the  assignment  of  average  gradients 
from  the  surface  to  the  two  upper  planes  in  the  free  air.  The 
temperatures  were  computed  on  these  planes  for  enough  sta- 
tions to  permit  drawing  systems  of .  isotherms  with  accuracy. 
As  regards  the  3,500-foot  plane,  the  temperatures  were  found 
from  the  free  air  gradients  for  stations  outside  the  plateau  and 
of  lower  elevation  than  3,500  feet;  for  points  within  the 
plateau  the  temperatures  on  that  plane  were  taken  from  the 
diagrams  of  vertical  temperatures,  previously  constructed; 
these  two  systems  agree  well  together,  and  the  isotherms  are 
continuous.  The  isotherms  on  the  10,000-foot  plane  are 
simple  curves  joining  the  Atlantic  and  Pacific  districts  and 
present  no  trouble  in  crossing  the  plateau.  There  is  one  re- 
sult of  interest,  however,  at  the  surface  of  the  plateau,  which 
I  call  "gradient  refraction. "  Within  the  plateau  the  vertical 
temperature  gradient  averages  about  0. 195°  per  100  feet,  and 
in  the  free  air  for  the  eastern  districts  about  0.300°  up  to 
10,000  feet.  Now  it  is  evident  that  this  plane  is  high  enough 
above  the  plateau  to  escape  the  influence  of  the  surface  con- 


ditions, and  that  it  is  in  the  midst  of  the  rapidly  drifting  cur- 
rent of  air  whose  direction  is  eastward,  so  that  quite  uniform 
temperature  must  prevail  along  the  same  parallel  of  latitude. 
Hence,  it  follows  that  by  using  the  smaller  gradient  0. 195°  to 
the  surface  of  the  plateau,  larger  values  than  0.300°  must  be 
employed  from  the  surface  to  10,000  feet,  if  the  average  gra- 
dient is  to  be  about  0.300°,  such  as  it  would  be  if  the  plateau 
were  removed.  Therefore  at  the  surface  of  the  plateau  there 
is  something  like  an  abrupt  change  in  the  gradients  which  is 
similar  to  refraction.  Finally,  by  means  of  the  temperatures 
thus  found  and  the  relative  humidities,  assumed  to  be  the 
same  as  for  the  surface  stations,  the  vapor  tension  on  the 
3,500-foot  plane  was  computed.  For  the  10, 000-foot  plane  it 
was  assumed  that  the  relative  humidity  is  50  per  cent  of  the 
surface  amount  at  all  places;  this  may  be  subject  to  criticism, 
but  it  is  near  the  truth  and  the  effect  on  the  vapor  tension  of 
even  considerable  changes  in  the  relative  humidity  would  be 
unimportant  at  the  low  temperatures  prevailing  at  that  altitude. 

THE  FIRST  COMPUTATION  OF  BI    Bf 

Instead  of  computing  the  values  of  /,,  <>v  and  /s,  ct,  for  the 
several  stations  at  the  outset,  the  work  was  much  shortened  by 
interpolating  the  values  of  all  this  data  on  selected  points  of 
the  charts,  namely,  centers  of  reduction;  that  is,  where  the 
meridians  5°  apart,  125°,  120° 65°,  cross  the  parallels 


5°  apart,  55°,  50C 


30° 


On  these  centers  of  reduction 


the  sea  level  Bm,  to,  ea  were  also  drawn  from  the  charts,  so 
that  the  data  is  complete  for  reducing  the  sea-level  pressures 
to  the  higher  planes.  There  are  two  objects  gained  by  this 
method  of  discussion;  (1)  the  work  of  computation  is  short- 
ened very  much;  and  also  (2)  the  result  affords  an  admirable 
check  on  the  entire  system  of  reductions,  as  will  be  seen  by 
what  follows.  The  pressures  /?,  and  B}  on  the  3, 500-foot  plane 
and  the  10,000-foot  plane,  respectively,  were  computed  by  the 
logarithmic  tables  from  the  data  thus  obtained  on  the  (tenters  of 
reduction,  and  the  corresponding  systems  of  isobars  were  drawn. 
There  now  exists  the  same  general  harmony  in  these  isobars 
as  on  the  sea-level  plane,  and  no  further  corrections  are  re- 
quired. It  is  to  be  especially  noted  that  in  the  plateau  region 
the  reductions  from  sea  level  to  the  upper  planes  were  made  by 
the  same  principles  as  if  it  had  b.eeu  a  free  air  column,  so  that 
all  plateau  questions  are  laid  aside. 

THE  SECOND  COMPUTATION  OF  BJt    B}  .       » 

From  the  B^  and  Bt  charts  the  pressures  belonging  to  all  the 
stations  were  interpolated,  so  that  the  values  of  /',,  /^,  to  be 
derived  by  a  direct  computation  from  the  station  data  could  be 
compared  as  a  check.  Meanwhile  the  several  station  reduction 
tables  to  the  three  planes  had  been  completed,  and  as  a  final 
check  the  three  values,  Ba,  Bv  Bv  were  computed  and  com- 
pared with  the  values  derived  from  the  charts,  as  explained  in 
the  first  process.  The  differences  between  the  two  sets  of  val- 
ues for  Ba,  Bt,  /?,  were  about  the  same  on  the  three  planes; 
they  average  about  0.010  inch,  the  majority  being  0.000  or 
0.010  inch,  a  few  0.020  inch,  with  occasional  larger  variations 
due  to  errors  of  computation  readily  detected,  or  to  a  local 
peculiarity,  involving  a  slight  readjustment  of  the  corrections 
in  the  station  tables.  These  checks,  therefore,  involved  the 
three  distinct  parts  of  the  entire  discussion,  since  the  process 
lias  been  arranged  practically  in  a  circuit  so  as  to  pass  from  the 
station  Bn  to  Iit  and  Bt  by  two  separate  routes,  as  described. 
Hence,  ( 1 )  the  processes  of  eliminating  the  plateau  effect,  and 
of  computing  the  temperature  arguments  /  and  fl  were  success- 
ful; (2)  the  logarithmic  tables  and  the  numerical  station 
tables  are  in  agreement;  (3)  the  charts  are  accurately  drawn, 
and  represent  the  observations  with  precision. 

As  the  result  of  this  discussion  we  have  prepared  charts  for 
the  United  States  and  Canada,  giving  the  monthly  and  annual 
normals  of  pressure,  temperature,  and  vapor  tension  on  the 
sea-level  plane,  the  3,500-foot  plane,  and  the  10,000-foot  plane. 


also  the  relative  humidity  on  the  sea-level  plane,  i.  e.,  130 
charts  for  these  data.  There  are  also  charts  of  gradients  of 
temperature  in  latitude,  in  longitude,  and  in  altitude;  and 
charts  of  pressure  variations  for  a  few  selected  hours  referred 
to  the  mean  of  24  hourly  observations.  Furthermore,  the 
corresponding  numerical  values  are  entered  in  a  summary 
table  for  all  stations  on  the  sea-level  plane,  about  265  in 
number;  also  for  all  the  stations  which  Were  in  use  by  the 
Weather  Bureau,  either  in  the  United  States,  Canada,  and  i 
the  West  Indies,  at  the  beginning  of  the  year  1900,  or  which  ' 
have  been  opened  for  service  since  that  date,  making  about 
175  on  the  upper  planes. 

It  has  not  been  found  necessary  to  revise  any  of  the  reduc- 
tions to  sea  level  since  the  tables  were  put  in  operation  on 
January  1,  1902,  showing  that  they  bear  the  test  of  practical  j 
work  at  the  hands  of  many  observers.      The  station  tables  for 


the  upper  planes  will  soon  be  tried,  and  an  estimate  made  as 
to  their  value  in  increasing  the  accuracy  of  the  forecast  sys- 
tem of  the  Weather  Bureau. 

We  conclude  with  the  remark  that  the  pressure  observations 
and  computations  of  the  United  States  have  been  at  last  placed 
upon  a  strictly  scientific  basis,  and  that  all  the  corrections  re- 
quired by  theory  will  be  systematically  applied  in  the  future, 
and  the  entire  series  from  1873  onwards  will  be  kept  strictly 
homogeneous.  We  shall,  therefore,  for  the  first  time  be  ready 
to  take  up  the  problems  of  seasonal  variation  of  the  weather, 
the  changes  of  the  climate  and  crop  from  year  to  year,  and 
also  the  true  cosmical  problems  involved  in  the  radiation  effects 
of  the  sun  upon  the  earth's  atmosphere.  Even  if  we  do  not 
ourselves  succeed  in  resolving  these  questions,  we  shall  have 
left  this  portion  of  the  data  in  form  for  others  to  make  reliable 
discussions. 


II.— METHOD    OF    OBSERVING    AND    DISCUSSING    THE    MOTIONS    OF    THE    ATMOSPHERE.' 


INTRODUCTORY    REMARKS. 

It  has  been  suggested  to  me  that  it  would  be  advantageous  to 
many  who  are  interested  in  the  progress  of  modern  meteor- 
ology, if  the  results  of  the  observations  on  clouds,  which  were 
made  by  the  United  States  Weather  Bureau  during  the  years 
1896-97,  could  be  put  in  a  more  compact  form  than  was  adopted 
in  the  original  report.2 

I  am  the  more  inclined  to  present  anew  some  of  my  results 
because  of  the  extensive  use  that  has  been  made  of  the 
American  observations  generally  in  Dr.  J.  Hann's  Lehrbuch 
der  Meteorologie,  1901.  In  this  judicious  and  comprehensive 
summary  of  the  state  of  meteorology  at  the  end  of  the  nine- 
teenth century,  Dr.  Hann  has  given  very  generous  recognition 
to  the  contributions  of  the  United  States,  including  the  Weather 
Bureau  and  the  Blue  Hill  Observatory,  to  the  advancement  of 
meteorology.  But  more  important  than  this,  the  views  therein 
adopted  regarding  the  theories  of  the  circulation  of  the  atmos- 
phere in  the  general  and  the  local  cyclones,  are  fully  in  accord 
with  the  ideas  set  forth  in  my  report  on  the  cloud  observations 
of  1896-97.  It  is  apparent  that  meteorology  is  at  last  secur- 
ing a  set  of  principles. founded  on  observations,  which  will  super- 
sede much  that  has  been  heretofore  taught  in  this  connection. 
It  is  therefore  important  to  explain  the  results  of  the  Weather 
Bureau  observations  of  1896—97  as  briefly  and  simply  as  possible. 

Taking  a  very  general  view  of  the  present  state  of  meteor- 
ology, it  may  be  proper  to  classify  the  conditions  as  follows: 
The  statistical  side  of  the  subject  is  being  rapidly  worked  up,  so 
that  our  knowledge  of  facts  is  relatively  quite  complete  in  cli- 
matology, and  in  the  diurnal  and  annual  periods  of  the  various 
atmospheric  elements,  namely,  pressure,  temperature,  vapor 
tension,  and  wind  direction,  in  different  parts  of  the  world, 
so  far  as  they  prevail  in  the  strata  near  the  ground.  But  in 
the  upper  strata  our  knowledge  of  these  elements  is  still  very 
limited,  though  it  has  been  considerably  extended  during  the 
past  ten  years,  by  the  cloud  observations,  and  the  balloon  and 
kite. ascensions.  On  the  theoretical  side  of  static  meteorology 
it  may  be  said  that  meteorological  analysis  is  well  advanced, 
as  far  as  concerns  the  barometric  relations  of  pressure  to  the 
height,  the  temperature  and  vapor  tension  variations,  and  the 
adiabatic  thermodynamics  generally.  The  practical  extension 
and  application  of  these  formula}  to  the  upper  strata  is  making 
fair  progress,  and  is  likely  to  result  in  very  definite  knowledge 
of  the  true  state  of  the  atmosphere  throughout  its  extent.  In 
dynamic  meteorology,  however,  that  is  in  the  hydrodynamics 
of  the  atmosphere,  affairs  are  in  an  unsatisfactory  condition, 
and  they  can  be  reclaimed  only  by  pursuing  a  sound  policy 
regarding  them.  Looking  over  the  entire  field,  one  is  sur- 
prised to  find  that  but  little  has  been  done  in  the  preliminary 
and  the  most  necessary  stages  of  this  work,  in  order  to  make 
the  dynamics  of  the  atmosphere  a  practical  scientific  problem. 
It  is  wasting  time  to  speculate  on  the  mathematical  analysis 
of  the  motions  of  the  atmosphere  till  we  know  what  the  motions 
are,  simply  as  a  case  of  kinematics.  In  other  words,  the  paths 
of  motion  of  the  average  air  currents  should  be  systematically 
worked  up  all  over  the  world,  as  the  indispensable  prelimi- 
nary to  this  study.  Of  course  the  obstacle  in  the  way  of  doing 
this  is  the  invisibility  of  the  air  itself,  and  the  labor  of  making 
any  observations  on  its  direction  and  velocity  of  motion  much 


1  Reprinted  from  the  Monthly  Weather  Review  for  February,  1902. 

2  Report  on  the  International  Cloud  Observations,  May  1  1896,  to  July 
1,  1897,  by  Prof.  Frank  H.  Bigelow.     Report  of  the  Chief  of  the  Weather 
Bureau,   1898-99,  Vol.  II. 

F.  H.  B. 2 


I  above  the  ground.  It  was  for  supplying  just  this  need  that 
the  international  cloud  observations  of  1896-97  were  instituted, 
and  to  it  they  have  contributed  a  valuable  amount  of  data. 

Furthermore,  there  are  the  great  physical  problems  con- 
nected with  the  absorption  of  the  sun's  radiant  energy  in  the 
atmosphere,  its  separation  into  several  kinds  of  energy — elec- 
tric and  magnetic  energy,  heat  energy  of  the  visible  and  invisi- 
ble spectrum,  and  so  on.  Also  there  is  the  question  to  be 
answered  as  to  the  amount  of  the  solar  output  itself,  the  varia- 
tions from  its  mean  value,  how  much  and  what  kinds  of  energy 
are  absorbed  in  the  upper  strata,  and  what  in  the  lower.  The 
circulation  of  the  atmosphere  in  its  details  really  goes  back  to 
these  questions  about  which  we  know  only  a  very  little.  Hence, 
in  a  word,  the  deficiency  of  modern  meteorology  is  in  the 
dynamics  of  the  upper  and  middle  strata  of  the  atmosphere. 

NOTATION    AND    COORDINATES. 

It  is  not  an  exceptional  fact  in  the  history  of  science  that 
in  the  first  stages  of  its  development  meteorology  should  have 
grown  up  in  a  rather  haphazard  fashion,  especially  as  it  was 
dealing  with  a  subject  of  popular  interest,  wherein  many  ob- 
servers were  concerned  in  getting  observations  of  one  kind  or 
another  without  much  regard  to  their  ultimate  use  in  mathe- 
matical analysis.  As  the  result  of  this  lack  of  purpose  the 
confusion  became  so  great  between  the  methods  of  observing 
and  recording  in  different  parts  of  the  world  that  when  com- 
parative studies  were  begun  the  difficulties  arising  from  the 
want  of  homogeneity  were  seriously  felt.  In  order  to  remedy 
this  state  of  confusion  the  International  Meteorological  Com- 
mittee have  been  laboring  for  years  to  introduce  uniformity 
into  the  methods  adopted  by  meteorologists.  Much  has  al- 
ready been  accomplished,  and  yet  there  are  at  least  two  very 
important  steps  that  remain  to  be  taken.  The  first  is  to  use 
only  one  system  of  measures,  as  the  metric  in  place  of  the  met- 
ric and  the  English ;  and  the  second  is  to  conduct  and  discuss 
|  the  observations  in  such  a  way  that  in  their  published  form 
they  shall  be  in  perfect  order  to  meet  the  requirements  of  the 
fundamental  mathematical  equations,  either  static  or  dynamic, 
as  they  are  needed.  At  present  meteorological  observations 
are  about  evenly  divided  between  the  English  and  the  metric 
systems  of  measures,  since  the  former  is  in  use  in  Great 
Britain,  Canada,  United  States,  south  Africa,  Australia,  and 
India ;  while  the  latter  prevails  in  Europe,  Asia  generally, 
Japan,  north  Africa,  and  South  America.  Thus  it  is  necessary 
to  translate  the  figures  from  one  system  to  the  other  in  prepar- 
ing the  data  for  the  world  and  for  cosmical  problems;  also  two 
sets  of  reduction  tables  are  required  for  all  the  elements,  and 
two  sets  of  constants  in  all  the  formulae.  The  more  lamentable 
defect  occurs  from  the  fact  that  the  observations  are  made 
without  relation  to  their  final  use  in  mathematical  discussions 
involving  the  motions  of  the  atmosphere.  Indeed  almost  no  th- 
ing has  been  done  to  give  us  the  true  vector  components  of 
motion  in  the  observations,  so  that  they  shall  be  in  form  for 
immediate  introduction  into  the  equations.  The  standard 
equations  have  been  presented  by  different  authors  in  many 
equivalent  forms,  and  in  consequence  the  subject  has  been 
made  unnecessarily  complex  and  difficult  for  students.  The 
entire  body  of  fundamental  equations  in  meteorology  is  not 
very  large,  but  the  amount  appears  to  be  much  greater  than 
it  really  is  by  reason  of  the  manifold  notations  and  symbols 
which  have  been  employed.  No  more  valuable  reform  could 
be  instituted  than  that  of  causing  the  same  physical  quantity 

9 


10 


to  be  always  represented  by  the  same  symbol.  Thus,  for  ex- 
ample, barometric  pressure  />',  pressure  in  units  of  force  P, 
pressure  in  units  of  weight  />,  would  put  us  in  harmony  with 
the  leading  works  in  hydrodynamics  and  thermodynamics; 
then  absolute  temperature  T,  thermometric  temperature  /, 
mean  temperature  of  the  air  column  l>,  vapor  tension  e,  maxi- 
mum vapor  tension  E,  absolute  weight  of  vapor  //,  weight  of 
the  unit  volume  a,  specific  weight  /•',  and  relative  humidity 
R.H.  For  rectangular  coordinates,  displacements  (JT,  y,  z)  », 
velocities  (it,  r,  ir)  (/,  accelerations  (u,  v,  it)f,  angular  velocities 
((u,,  u>f  <»,);  for  cylindrical  coordinates,  radius  ra,  angle  about 
axis  of  rotation  y,  for  polar  coordinates,  radius  vector  r,  polar 
distance  0,  angle  about  axis  of  rotation  /. 

I  have  felt  the  weight  of  these  considerations  in  my  com- 
parative studies  so  much  that  special  pains  have  been  taken 
in  my  report  to  exhibit  all  the  fundamental  equations  in  a 
standard  system  of  notation,  and  also  to  reduce  the  analyses 
of  several  authors  to  the  same  standard  system  for  the  sake  of 
ready  intercomparison.  Also,  as  it  seems  to  be  of  the  utmost 
importance  that  the  observations  taken  to  determine  the 
motions  of  the  atmosphere  should  be  made  in  a  form  appro- 
priate for  use  in  the  dynamic  equations  without  further  trans- 
formation of  the  data,  particular  care  was  taken  to  make  the 
cloud  observations  conform  to  these  requirements.  It  was  not 
possible  to  bring  about  this  harmony  between  observations  and 
the  analytical  theory  without  introducing  some  radical  changes 
in  the  methods  heretofore  followed  by  meteorologists,  both  in 
the  conduct  of  the  observations  and  in  the  analytic  develop- 
ment of  the  equations.  Accordingly,  some  account  of  these 
changes,  as  well  as  of.  the  new  results  which  were  deduced 
from  the  cloud  observations  made  by  the  United  States  Weather 
Bureau  in  1896-97,  will  be  given  in  the  following  pages. 

THE    AXES    OF    COORDINATES. 

The  first  decision  that  must  be  made  in  establishing  a  funda- 
mental system  of  notation  has  regard  to  the  choice  of  the  axes 
of  coordinates  which  shall  be  placed  at  the  base  of  the  entire 
study,  since  all  the  algebraic  signs  of  the  quantities  depend- 
ing on  the  observations  which  are  to  be  substituted  in  the 
equations,  must  be  determined  from  the  adopted  positive 
direction  of  the  axes.  This  choice  depends  practically  upon 
two  facts,  (1)  that  the  radius  of  the  earth  is  drawn  positive 
outwards,  since  r  increases  from  the  center,  and  (2)  that  the 
right-hand  rotation  is  adopted  generally  in  modern  scientific 
researches.  It  is  true  that  some  of  the  German  mathemati- 
cians use  the  left-hand  rotation,  but  the  trend  is  toward  a 
universal  adoption  of  the  right-hand  system.  If  we  take  as 
the  primary  radius  that  of  the  earth's  axis  of  rotation  in  the 
Northern  Hemisphere,  as  is  most  appropriate  for  all  scientific 
problems  except  in  terrestial  magnetism,  then  in  polar  co- 
ordinates the  positive  angular  development  in  polar  distance, 
0,  is  southward ;  next,  with  the  right-hand  rotation  about  this 
axis  extended  upward,  the  positive  development  of  ).,  the  angle 
in  longitude,  is  eastward.  Hence,  for  all  systems  of  coordi- 
nates, polar,  cylindrical,  and  rectangular,  the  azimuth  rotation 
is  from  the  south  through  the  eaxt,  north,  and  west.  Unfortunately 
this  is  in  the  opposite  direction  to  the  azimuth  rotation  adopted 
in  astronomy,  in  navigation,  and  in  popular  meteorology,  be- 
cause in  these  branches  of  science  the  simple  practical  consid- 
eration has  been  to  follow  the  sun  in  its  diurnal  course,  so  that 
azimuth  circles  and  compass  cards  are  numbered  around  in  the 
clockwise  or  left-hand  rotation.  For  many  statistical  pur- 
poses, as  where  average  wind  directions  are  to  be  computed, 
it  makes  little  difference  what  system  of  notation  is  used, 
because  these  data  do  not  look  beyond  their  own  immediate 
purposes.  But  where  we  have  to  deal  with  a  system  of  equa- 
tions it  is  not  so.  If  we  take  the  first  set  of  equations, 
International  Cloud  Report  (154),  for  linear  velocities  due  to 
rotation, 


i1,  =  '•  —  s  '",  -I-  •'• »'., 

JOj   =  W  —  X  w.t  -f    //  1,1 , 

where  .r,  »/,  z  are  linear  distances,  u,  i;  u-  are  linear  velocities. 
,  <at,  n>.t  are  angular  velocities,  it  is  evidently  necessary  that 
all  these  should  be  defined  most  carefully.  According  to  the 
statement  given  above, 

+  (./-,  (/)  are  referred  to  the  southward  axis, 
+  '(.'/>    '')  are  referred  to  the  eastward  axis, 
+  (z,   ir)  are  referred  to  the  zenith  ward  axis, 
but  everything  will  go  wrong  if  <«3  is  not  taken  to  rotate  posi- 
tively about  the  axis  3,  from  the  south  through  the  east,  instead 
of  through  the  west.     Hence,  </*,  turns  the  axis  of  y  to  z,  <«.,  the 
axis  of  z  to  .r,  <«3  the  axis  .r  to  ;/,  in  cyclical  order.      Thus  it  hap- 
pens that,  having  adopted  this  system  of  rotation,  it  was  found 
necessary  to  transform  the  equations  of  some  well-known  mathe- 
matical papers  in  dynamic  meteorology  to  agree  with  it. 

THE    AZIMUTH    ROTATION. 

There  is  one  further  difficulty  to  overcome  in  regard  to  the 
popular  meteorological  system.  An  observer  determines  the 
direction  of  the  wind  by  looking  toward  it  and  feeling  the  force 
of  it  on  his  face,  or  he  sets  up  a  wind  vane  with  an  arrow  point- 
ing in  the  direction  from  which  it  blows.  This  is,  however,  ex- 
actly contrary  to  the  method  that  mathematical  physicists  em- 
ploy, for  they  describe  a  stream  line  by  the  direction  luifiril 
which  it  flows.  An  arrow  is  drawn  on  a  map  "  down  stream  " 
to  show  how  the  current  Hows,  or  on  the  weather  map  an  arrow 
is  said  to  "fly  with  the  wind. ' '  In  the  latter  case  meteorolo- 
gists are  inconsistent  with  themselves,  but  they  adopt  the  cor- 
rect principle  in  their  precept  on  the  map.  If  we  describe  a 
wind  as  having  a  velocity  of  so  many  miles  per  hour  from  a 
given  direction,  say  the  north,  this  must  be  changed  in  azimuth 
through  180°  to  the  south  in  order  to  be  of  use  in  analytic 
work.  We  have  thus  to  make  two  mr/vw/N  in  the  common 
meteorological  system:  (1)  the  wind  vector  must  be  turned 
through  180°,  and  (2)  the  azimuth  must  be  numbered  from 
the  south  =  0°  toward  the  east  =  90°,  north  =  180°,  and  west 
=  270°.  These  two  changes  render  it  impossible  to  use  the 
ordinary  wind  records  which  are  found  in  meteorological  re- 
ports without  making  this  transformation.  While  it  is  not  a 
very  important  matter  for  a  few  individual  cases,  it  becomes  a 
serious  task  to  do  the  work  of  remodeling  the  figures  for  a 
large  amount  of  data.  It  was  for  the  purpose  of  saving  this 
labor  that  the  observations  of  the  Weather  Bureau  were  exe- 
cuted in  1896-97  on  the  correct  system,  so  that  all  the  figures 
appearing  in  the  final  report  should  be  at  once  ready  for  use 
in  the  equations  of  hydrodynamics. 

In  order  to  facilitate  the  understanding  of  the  discussion 
that  follows,  a  chart  is  introduced  to  show  the  scheme  of  the 
operations.  Fig.  1  gives  a  comparison  of  the  azimuth  system 
commonly  employed  by  meteorologists  with  the  one  adopted 
in  this  report.  The  former  is  the  left-hand  rotation,  such  as 
astronomers  use,  and  is  counted  from  the  south  through  the 
west  point  of  the  horizon.  In  this  system,  also,  the  observer 
faces  the  wind  and  gives  the  azimuth  of  the  direction  J'm/n 
•triiirh  it  blows.  This  has  been  abandoned  for  the  reasons  al- 
ready mentioned,  and  in  place  of  it  is  substituted  the  right- 
hand  rotation,  wherein  the  azimuth  is  counted  from  the  south 
through  the  east  point.  Here  the  observer  receives  the  wind 
on  his  back  and  looks  toward  the  direction  in  which  the  arrow 
flies  and  toward  whifh  the  air  is  moving.  Thus  a  wind  from 
tin-  .Yirby  the  former  system,  with  azimuth  angle  135°,  be- 
comes a  wind  toirarif  ///<•  XA'.  with  the  azimuth  angle  45°,  by 
the  latter  system.  The  working  out  of  the  results  by  this  sys- 
tem, ready  for  analytic  discussion,  as  will  be  seen,  involves  a 
minimum  of  computation,  and  besides  this  it  reduces  all  the 
velocity  components  to  the  fundamental  rectangular  system  of 


11 


coordinates  adopted  by  Ferrel  in  his  treatise,  and  continued 
hi  iny  "  Standard  System  "  which  forms  a  part  of  the  Report. 


Ft<;.  1. — Comparison  of  the  two  azimuth  systems. 

Left-hand  rotation  gives  azimuth  of  "motion  from." 
Kight-liaiid  rotation  gives  azimuth  of  "motion  toward. " 
Positive  translation  i-  vertically  upward. 

The  Marvin  uephoscopes,  with  which  the  observations  were 
made,  were  all  graduated  to  read  in  a  right-hand  (anticlock- 
wise )  azimuth ;  the  theodolites  were  also  read  in  the  same  direc- 
tion, and  the  azimuths  of  Tables  (>  and  9  of  the  International 
Cloud  Report  are,  therefore,  in  accord  with  the  coordinate 
directions  of  the  formula?  which  are  developed  in  the  following 
portions  of  the  same  Report. 

THE    COMPOSITION    AND    RESOLUTION    OF    THE    VECTORS    OF    MOTION. 

It  is  sometimes  necessary  to  construct  the  resultant  velocity 
and  direction  of  the  motion  of  the  air  at  a  given  place  out  of 
a  large  number  of  individual  observations,  as  in  forming  charts 
20-35,  International  Cloud  Report,  for  example,  and  the  fol- 
lowing practical  devices  were  found  convenient.  Suppose  it 
is  desired  to  determine  the  velocity  aud  direction  of  motion  in 
the  cumulus  cloud  level  on  all  sides  of  a  low  area,  as  in  the 
Mississippi  Valley.  \Ve  can  best  proceed  as  follows:  Take  a 
piece  of  tracing  paper,  and  select  a  large  number  of  cloud  maps 
showing  about  the  same  configuration  of  the  isobars,  so  that 
the  centers  of  the  cyclones  are  located  in  a  given  district.  Then 
lay  the  paper  on  the  cloud  maps  in  succession,  and  trace  the 
arrows  showing  the  cloud  motion  wherever  an  observation  is 
found  on  the  map.  Mark  the  center  on  the  first  map  and  pre- 
serve it  so  as  to  place  it  in  coincidence  with  the  other  cyclonic 
centers.  Continue  to  fill  the  paper  till  some  such  composite  of 
arrows  is  obtained  as  is  shown  within  the  square  of  fig.  2.  A 
scale  map  of  squares,  or  any  other  adopted  division  of  areas, 
is  to  be  prepared  as  large  as  the  tracing  paper,  and  the  two 
are  placed  together  so  that  the  scale  diagram  marks  oft'  the 
arrows  of  the  composite  map  into  groups,  within  each  of  which 
it  is  proposed  to  find  the  resultant.  Then  covint  out  the  num- 
ber of  arrows  pointing  N,  NE,  E,  etc.,  in  succession  for  eight 
directions,  giving  in  our  example,  N=4,  NE=4,  E=10,  etc.; 
take  the  excess  in  four  directions,  as  S  =  7,  E  =  (>,  SE  =  7, 
SAV  =  10;  plot  these  results  on  a  diagram  and  resolve  SE=  7 
into  E  =  5  and  S  =  5 ;  also  S\V  =  10  into  W  =  —  7  and  S  = 
+  7;  make  the  sums  E=  +4  and  S=  +19;  plot  these  compo- 


nents and  obtain  the  resultant  I"=20;  the  angle  <p =9°  can  be 
found  by  the  use  of  a  circular  protractor,  or  it  can  be  computed 

-p 
by  the  formula  tan  <?=    -,  having  regard  to  a  change  of  alge- 

S 

braic  signs  for  \V  =  —  E  and  N  =  —  S. 


Wind  vector. 


S=7  . 


SE=:7  .  . 
SW=10. 


Component. 


+6 

+5 
—7 


+5 

+7 


+  19 


V=  20;  <f  =  9°,  that  is,  =  S  9°  E. 
tan  ,=1=4=0.21. 


Fig.  2. —Example  of  the  graphic  composition  of  wind  vectors. 
It  is  easy  to  perform  a  large  amount  of  graphic  composition 
in  a  short  time  after  a  little  practise,  by  arranging  this  work 
systematically.  In  the  collection  of  vectors  from  the  maps  the 
total  number  will  differ  from  square  to  square,  and  it  is  nec- 
essary to  reduce  the  resultant  to  a  common  standard  number. 
Suppose  we  adopt  40  arrows  as  the  standard,  then  the  com- 
pleted resultant  velocity  must  be  reduced  in  that  proportion. 

40 
Our  example  contained  G4  arrows,  hence  20  x    . .  =  12,  and  12 

is  to  be  adopted  as  the  average  velocity  of  the  motion  in  the 
azimuth  9°,  that  is,  S  9°  E.  These  resultants  assume  that  the 
average  of  a  number  of  observed  directions  gives  a  rdutire 
velocity  of  motion,  which  can  be  reduced  to  an  absolute  velocity 
as  soon  as  the  true  mean  motion  is  determined  from  some  other 


12 


source,  as  by  theodolite  observations.  Charts  constructed  ill 
this  way  are  quite  correct  as  to  the  direction  of  the  motion  of 
the  atmosphere,  and  they  give  the  relative  velocities  in  differ- 
ent parts  of  a  cyclone  or  anticyclone  with  sufficient  precision 
to  permit  further  important  studies. 

By  the  neplioscope  observations  the  actual  velocities  in 
different  portions  of  the  area  surrounding  the  center  of  motion 
were  computed  and  collected  for  the  several  subareas  about 
the  highs  and  lows,  as  explained  in  Chapter  7  of  the  Inter- 
national Cloud  Report.  The  relative  velocities  there  recorded 
can  be  turned  into  actual  velocities  by  utilizing  the  corre- 
sponding theodolite  observations.  The  nephoscope  refers  all 
the  observed  motions  to  the  1000-meter  plane,  and  the  theodo- 
lite to  the  actual  plane  of  motion  at  the  height  given  by  the 
angular  measurements.  I  note  that  Dr.  J.  Haun,  in  his  Lehr- 
buch  der  Meteorologie,  pages  272,  273,  and  275,  attributes 
certain  cloud  heights  of  the  Weather  Bureau  observations  to  the 
nephoscopes,  but  this  is  a  mistake,  because  all  our  heights 
were  determined  by  the  theodolites.  The  heights  associated 
with  the  nephoscope  observations  were  adopted  for  translating 
the  relative  velocities  of  the  nephoscopes  into  acttial  velocities, 
and  his  impression  doubtless  arose  from  printing  such  adopted 
heights  in  conjunction  with  the  other  data  which  were  derived 
from  nephoscopes. 

THE    RESOLUTION    OF    FORCES. 

In  the  study  of  the  motions  of  the  atmosphere  at  all  levels 
there  are  two  types  of  resolution  of  vectors  to  be  provided  for 
in  the  discussion,  the  first  in  rectangular  coordinates, 
+  x  =  South,  +  y  =  East,     +  z  =  zenithward, 
—  x  =  North,  —  y  =  West,  —  z  =  nadirward, 
in  order  to  apply  them  to  the  motions  of  the  general  circula- 
tion over  the  entire  hemisphere ;  and  the  second  in  cylindrical 
coordinates, 
+  x  =  radial  outward, 

—  x  =  radial  inward, 

-(-  y  =  tangential  counter-clockwise, 

—  y  =  tangential  clockwise, 

4.  2  =  vertical  upwards  on  the  axis, 

—  z  =  vertical  downwards  on  the  axis,  the  results  being  used 
in  the   analysis  of  cyclones  and  anticyclones,  that  is  in  the 
local  circulations.      It  is  evident  that  the  vectors  provided  by 
the  theodolite  and  nephoscope  observations,  in  the  form  V  = 
velocity  and  <p  =  azimuth  counted  from  the  south  through  the 
east,  are  ready   for  simple  trigonometric  resolution  into  the 
velocity  coordinates  («,,  i>,)  in  the  four  quadrants  by  using  the 
proper  signs.     When  all  the  vectors  ( V,  <f)  are  resolved  in  the 
north-south  and   west-east  directions,  we   can  take  the  mean 
values  by  summation  and  then  compute  the  average  motion  of 
the  entire  mass  of  air  circulating  at  a  given  altitude  over  any 
locality.     It  is  necessary  to  obtain  these  mean  directions  of 
motion  for  the  entire  circulation,  in  order  to  be  able  to  resolve 
out  the  special  components  of  local  circulation   belonging  to 
the  cyclones  and  anticyclones.      In  the  report  on  the  Inter- 
national Cloud   Observations  the   results  of  this  rectangular 
resolution  of  the  observed  mean  vectors  are  set  forth  in  Table 
33  as  a  summary;  in  Tables  34,  35,  and  36  for  high  areas  in 
the  northern  and  southern  portions  of  the  United  States;  in 
Tables  38,  39,  and  40  for  the  low  areas  in  the  northern  and 
the  southern  portions  of  the  United  States;  in  Tables  42  and 
43  the  component  southward  and  eastward  velocities  in  highs 
and  lows,  also  in  selected  areas;  and  in  Tables  48,  49,  50,  and 
51  the  seasonal  velocities  in  the  upper  and  lower  cloud  levels. 
This   data  bears   directly  upon   the  problem  of  the  upper  air 
currents  in  the  general  circulation,  and  similar  data  ought  to 
be  obtained  in  all  portions  of  the  world. 

These  rectangular,  meridional,  and  longitudinal  velocity  com- 
ponents are  marked  w,,  r,  in  order  to  distinguish  them  from 


the  cylindrical  components  which  arc  designated  »,,  r,.  Hav- 
ing constructed  the  individual  components  of  general  motion 
",,  <\  in  all  the  subareas  together  with  the  corresponding 
normal  velocities,  it  is  evident  that  the  algebraic  differences 
between  them  gives  the  true  cyclonic  and  auticyclonic  compo- 
nents, still  in  rectangular  coordinates,  as  in  Tables  44  and  45. 
The  next  step  is  to  transform  them  into  cylindrical  coordinates 
ii.,,  ra  with  the  least  possible  labor.  For  this  purpose  it  is 
important  to  select  the  subareas  surrounding  the  center  of  a 
local  circulation  in  such  a  manner  as  will  contribute  to  that 
purpose.  If  the  areas  marked  out  by  the  5°  meridians  and 
parallels  of  latitude  are  taken,  it  is  impossible  to  transform 
the  component  velocities  without  a  most  tedious  computation. 
Such  areas  are  suitable  for  a  simple  display  of  the  stream  lines, 
but  they  do  not  readily  lend  themselves  to  the  composition 
and  resolution  of  vectors  of  motion. 

I  have,  therefore,  adopted  the  following  plan  for  the  sub- 
areas  surrounding  a  center  of  motion.  They  each  have  about 
the  same  relative  area,  and  they  are  distributed  us  far  as  pos- 
sible on  the  cardinal  lines  of  the  compass  direction.  They  are 
numbered  with  the  right-hand  rotation,  and  they  are  central 
each  on  some  cardinal  point  of  the  compass.  Thus  on  the 
north-south  line  17,  9,  3,  1,  5,  13  are  located;  on  the  west- 
east  line  19,  11,  4,  2,  7,  15  are  found;  while  18,  10,  6,  14  run 
from  NW  to  SE,  and  20,  12,  8,  16  from  SW  to  NE.  Hence 
it  is  seen  that  all  the  N-S  and  W-E  areas  are  immediately 
available  for  composition  in  rectangular  and  in  cylindrical 
coordinates,  while  the  rectangular  coordinates  of  the  NW-SE 
and  SW-NE  areas  which  come  from  the  first  collection  need 
only  a  simple  transformation  to  become  the  radial  and  tan- 
gential components  of  cyclonic  circulation.  These  latter  are 
to  be  ultimately  worked  out,  and  we  shall  then  have  three 
sets  of  coordinates  arranged  symmetrically  about  the  center 


270 


£90' 


Fio.  3. — Plan  of  the  subareas,  azimuths,  and  compass  points,  adopted 
in  high  and  low  areas,  for  the  discussion  of  cloud  observations. 

The  mean  of  areas  1  to  4=  I,  at  the  aveniK''  ilislnncc  2">0  kin. 
The  mean  of  areas  5  to  12=  II,  at  the  average  distance  75(1  km. 
The  mean  of  areas  13  to  20=111,  at  the  average  distance  1,J.">0  km. 

of  the  three  circles  I,  II,  III  at  certain  evenly  distributed  dis- 
tances. The  scale  of  the  original  diagram,  Chart  9,  Interna- 
tional Cloud  Report,  is  on  a  radius  of  3  centimeters;  on 
the  weather  maps  this  is  equivalent  to  15  centimeters,  where 
1  cetimeter  is  equal  to  100  kilometers.  The  adopted  scale 
is  therefore  one-fifth  the  scale  of  the  daily  weather  map, 
and  on  it  1  centimeter  represents  500  kilometers  or  310. 7 
miles,  and  one  millimeter  is  equivalent  to  50  kilometers,  or 
31.1  miles.  All  the  diagrams  of  the  Report,  as  far  as  possible, 
are  reproduced  on  this  scale,  but  they  are  readily  interpreted 
on  the  weather  map,  so  far  as  linear  dimensions  are  concerned. 

VECTORS  OF    MOTION  IN  HIGH  AND  LOW  AREAS HKrTAXH'LAR 

COORDINATE. 

In  order  to  prepare  the  observations  for  discussion  all  those 


13 


which  were  made  iii  the  same  sub  area  of  n  cyclone  or  an  auti- 
cvcloue  were  collected  together  in  each  cloud  stratum,  and 
tiic  resultant  of  all  these  individual  vectors  was  computed  in 
accordance  with  the  method  above  described.  The  individual 
observations  occur  iu  Table  9,  and  the  mode  of  collecting  them 
is  illustrated  in  Table  29,  page  363  of  the  Report.  For  con- 
venience, the  United  States  was  divided  into  six  districts:  1, 
Alberta;  2,  Lakes;  3,  New  England;  4,  Colorado;  5,  West 
Gulf;  C,  South  Atlantic;  so  as  to  arrive  at  a  conception  of  the 
prevailing  local  characteristics.  Hence  the  heading  of  the  form 
H — 2 — 15,  occurring  in  several  tables,  means  that  in  subarea 
15,  of  a  high  area  or  anticyclone  whose  center  is  iu  the  Lake  dis- 
trict, the  accompanying  observations  were  made  in  the  several 
cloud  strata,  and  also  at  the  surface  where  the  instrumental 
meteorological  data  are  given  at  the  three  daily  observations. 
Table  32  contains  the  resulting  vectors  V,.y>  for  the  northern 
and  southern  groups  by  districts,  also  for  the  four  seasonal 
quarters  of  the  year,  together  with  the  several  mean  values, 
all  this  extending  to  the  eight  cloud  strata.  In  this  table  the 
relative  velocities  are  given  as  derived  from  the  nephoscope, 
that  is  on  the  1,000-meter  plane. 

TAIILE  1. — Direction  and  velocity  of  motion  in  hiijh  and  low  areas — 
rectangular  coordinates.  * 


Compos 
Point 

Area 

number. 

i 
Cirrus;  average  height  9.  8  kilometers. 

High. 

IMVI. 

Ab.       <p           r 

.v«.     <f>  '      i' 

S 

1 

25      86     35.3 

1    111     51.0 

E 

2 

16      57     43.  1 

4      96     40.  2 

N    ; 

~3 

49      76     31.  4 

W 

4 

30     78     36.3 

u   lot    r,8.8 

S 

5 

37      66     37.  2 

50     97     31.4 

SE 

G 

20     67     32.  3 

33    101     44.1 

E 

NE 

N 

7 
8 
9 

23      90     58.8 
43      89     29  .  4 
51      81     34.  3 

10    109     36.3 

7      90     49.0 

NW 

10 

34    117     38.2 

3      92     27.4 

W 

11 

42      80     37.  2 

23      77     45.  1 

S\V 

12 

12    107     14.7 

6    105     56.8 

S 

13 

38      95     31.4 

27    123     33.  3 

SE 

14 

r,4     82     35.  3 

70     1)11     32.3 

E 

15 

28    \(X>    46.1 

49      85     34.  3 

NE 

16 

58      85    28.4 

24     90    30.  4 

N 

17 

27      88     36.3 

NW 

18 

25  .    96     33.  3 

2    122     39.2 

W 

19 

51      98     29.  4 

36      71     44.  1 

s\v 

•20 

63    109     30.4 

94      90     41.  2 

No.  of 
Mciin; 

obs  .... 
i    

736    
.     34.  9 

451 

'.     40.8 

*  Extracts  from  Tables  34  and  88.    • 

The  vectors  of  Table  32  are  plotted  on  Chart  13  of  the  Report, 
the  northern  in  red  and  the  southern  in  blue,  first  for  the  high 
areas  and  then  for  the  low  areas;  also  in  the  seasonal  groups  so 
that  the  comparative  motions  can  be  studied.  It  is  evident  that 
several  years  work  are  needed  to  produce  smooth  and  evenly 
balanced  mean  vectors,  which  shall  truly  represent  the  average 
circulation.  Espeoiullv  it  will  be  necessary  for  the  Canadian 
stations  to  cooperate  and  supply  the  vectors  wanting  in  the 
northern  subareas  of  our  three  northern  districts,  as  this  part 
of  the  circulation  usually  extends  into  Canada.  Furthermore, 
the  vectors  of  Table  32  are  collected  together  numerically  in 
Tables  34  to  40,  with  a  single  change,  namely,  that  the  ve- 
locities observed  on  the  1,000-nieter  plane  have  been  multiplied 


by  the  adopted  mean  height  of  the  given  cloud  stratum.  For 
example,  the  mean  height  of  the  cirrus  is  taken  as  9.8  kilometers, 
and  hence  the  mean  annual  velocity  l'=  3.6  of  the  cirrus  in 
high  area  No.  1.,  page  368,  is  multiplied  by  9.8,  and  it  is 
entered  at  the  beginning  of  Table  34  as  F=  35. 3.  I  have 
taken  the  cirrus  in  each  subarea  of  the  high  and  low  areas  to 
show  as  an  example  in  Table  1  and  fig.  4. 

HIGH.  LOW. 


FIG.  4.     (From  Chart  15.) 

These  vectors  are  plotted  on  Chart  15  (see  fig.  4),  which 
shows  the  annual  vectors  on  the  several  cloud  levels  in  high  and 
low  areas.  From  Chart  15  and  the  Tables  34-40  are  obtained 
the  data  for  discussing  the  mean  general  circulation  over  the 
United  States.  The  mean  total  velocities  in  high  and  low 
areas,  without  regard  to  direction,  are  found  by  taking  the  mean 
of  the  velocities  in  the  areas  of  Tables  (34-40)  and  they  are 
given  in  Table  33,  section  1,  as  in  the  following  example, 
Table  2: 
TABLE  2. — Total  velocitifs  in  high*  and  lows  without  regard  to  directions.* 


Clouds. 

High  areas. 

Low  areas. 

Height, 
Idiom. 

All 
groups. 

North- 
ern. 

South- 
ern. 

All 
groups. 

North- 
ern. 

South- 
ern. 

Ci 

9.8 
9.8 
8.1 
5.9 
4.5 
2.5 
1.5 
0.9 
0 

34.9 
39.1 
33.5 
30.  2 
23.5 
23.3 
11.2 
11.4 
4.8 

38.3 
42.6 
33.9 
31.1 
26.6 
22.7 
10.9 
12.2 
4.9 

30.4 
34.8 
30.5 
24.1 
19.7 
18.5 
10.4 
9.5 
4.8 
19 
36 

40.8 
39.8 
39.3 
36.0 
29.2 
28.6 
14.6 
11.1 
5.4 
15 
38 

44.6 
42.5 
43.8 
39.4 
32.6 
32.9 
17.4 
13.2 
5.3 

28.3 
36.3 
34.8 
30.5 
24.4 
21.1 
11.8 
8.6 
5.9 
28 
40 

Ci   S 

Ci    Cu 

A.  S 

A.  Cu          

S.  Cu      

Cu      

S     

Wind   

Range,  per  ct  . 
From  Table 

34 

35 

39 

'Extract  from  Table  :!:!,  Section  I. 

Table  2  shows  that  the  velocities  are  greater  in  the  north- 
ern circuit  than  in  the  southern,  and  greater  in  the  low  areas 
than  in  the  high  areas.  These  values  must  be  studied  in  con- 
nection with  the  barometric  gradients  to  form  a  theory  of  the 
dynamic  action  in  the  atmospheric  circulation. 

The  vectors  in  the  form  velocity  and  azimuth,  V,  <p,  are  next 
resolved  into  rectangular  components  in  the  north-south,  west- 
east  direction  by  the  trigonometric  rules,  and  the  results  are 
given  in  Tables  42,  43,  from  which  the  example  in  Table  3  is 
taken. 

The  means  of  these  components  are  taken  out  in  two  ways: 
(1)  the  algebraic  mean  which  gives  the  rectangular  coordinates 
of  motion  in  the  general  circulation  for  the  highs  and  the 
lows,  respectively.  The  mean  of  these  last  forms  the  normals 
from  which  the  true  cyclonic  components  are  computed,  and 
they  are  printed  in  heavy  faced  type.  These  results  are  col- 
lected together  in  Table  33,  Sections  II  and  III,  from  which 
the  extract  in  Table  4  is  made. 


14 


TABLE  3. — General  rectangular  components  of  motion  in  hiijli  ami  lon<  amis." 


(  irrus,  average  he- 

ght9.8kil»iiit>tiT!i. 

(  '.ml|.H^ 
|Mlillt. 

Area 

mi  lull.    I. 

High. 

Low. 

S  +             E  + 

S+              E  + 

S 

1 

+  2.5     +35.2 

—18.  3     +47.  7 

E 

•2 

+23.  5     +36.  2 

-  4.  2     +40.  0 

N 

3 

+  76     +30  5 

W 

4 

+  8.6     +35.5 

—14.  2     +57.  0 

s 

5 

+  15.1     +34.0 

-3.8     +31.2 

SE 

6 

+  12.6     +29.7 

-8.4     +43.3 

E 

7 

0.  0     +58.  8 

—11.8     +34.3 

NE 

g 

+  05     +29  4 

N 

9 

+  5.4     +33.9 

0.  0     +49.  0 

NW 

10 

—17.  3     +34.  0 

-  1.0     +27.4 

\V 

11 

+  6.4     +36.6 

+10.  1     +43.  9 

sw 

12 

-4.3     +14.1 

—14.  7     +54.  9 

s 

13 

-2.7     +31.3 

—18.1     +27.9 

SE 

14 

+  4.9     +34.9 

—  1.6     +32.3 

E 

15 

—  8.0     +45.4 

+  3.0     -1-34.2 

NE 

16 

+  2.5     +28.3 

0.  0     +30.  4 

N 

17 

+  13     +36  3 

NW 

18 

-  5.  2     +33.  1 

—20.  8     +33.  2 

W 

19 

-  4.  1     +29.  1 

+  14.4     +41.7 

SW 

20 

-9.9     +28.8 

0.  0     +41.  2 

Means 

+  1.97     +33.7 

—5.  26     +39.  4 

Normals 

—  1  6     +36.6 

-  1.6    +36.6 

Meaiis 

+0  66     +40  1 

—3  75     +32.7 

*  Extract  from  Tables  42  and  43. 

TABLE  4. — Southward  and  eastward  components  of  velocities  in  highs  and 

lows.* 


Clouds. 

High  areas. 

Low  areas. 

Means. 

Means. 

-fS  —  N 

+E  —  W 

+S  —  N 

—  5.26 
-  9.24 
—  3.00 
-  4.60 
—  2.38 
-  4.00 
-  0.  11 
-  1.32 
-  0.40 
43 

+E  —  W 

North. 

East. 

Ci  

+   1.97 
+   1.65 
—  0.60 
-  0.37 
-  0.07 
-  0.32 
-  0.13 
-  1.22 
-  0.69 
42 

+33.7 
+32.0 
+32.  6 
+27.2 
+22.1 
+  16.0 
+  5.1 
+  5.8 
+  1.1 
42 

+39.4 
+  35.9 
+37.2 
+31.3 
+24.3 
+24.3 
+  11.4 
+  7.8 
+  1.5 
43 

—  1.6 
-  3.8 
-  1.8 
-  2.5 
-  1.2 
-  2.2 
-  0.1 
-  1.3 
-  0.5 

+36.6 
+  34.0 
+34.9 
+29.2 
+23.2 
+20.2 
+  8.3 
+  6.8 
+  1.3 

Ci.  S    

Ci.  Cu 

A.  S 

A.  Cu  .    . 

S.  Cu  

Cu 

s 

Wind 

From  Table  . 

•Extract  from  Table :«,  Sections  II  and  III. 

The  most  important  remark  to  be  made  regarding  these  ex- 
tracts is  that  the  observations  show  an  average  northern  com- 
ponent in  the  United  States  in  all  levels,  provided  it  is  a  fact 
that  as  much  air  streams  through  the  low  areas  as  through  the 
high  areas  on  the  average.  (2)  The  subareas  were  collected 
into  two  groups,  those  having  a  southward  and  those  having 
a  northward  component.  Thus  we  have  a  southward  compo- 
nent in  high  areas  in  16,  8,  2,  7,  15,  6,  14,  and  in  low  areas  in 
18,  10,  4,  11,  19,  12,  20;  but  a  northward  component  in  high 
areas  in  18,  10,  4,  11,  19,  12,  20,  and  in  low  areas  in  16,  8, 
2,  7,  15,  6,  14.  The  means  from  these  groups  give  the  mean 


TABLE   5.  —  Component  velocities   in  selected  areas   between   high  and   low 


centers.  * 


Clouds. 

Selected  areas. 

Southward. 

H.   16,  8,  2,  7,  IB,  6,  14 
L.   18,10,4,11,  19,12,20 

Northward. 
L.  16,  8,  2,  7,  15,  6,  14 
H.  18,10,4,11,19,12,20 

Ci  

+    0.66 
-    2.11 

+  4.95 
+  2.79 
+  6.24 
+10.22 
+  6.52 
+  5.25 
+  2.23 

+40.  1 
+36.9 
+38.7 
+26.5 
+  23.7 
+22.1 
+  9.6 
+  7.5 
+  3.2 

-   3.75 

-  3.89 
-  7.34 
-  7.47 
-  7.78 
—  11.13 
-  8.13 
—  7.97 
-  3.25 

+32.7 
+38.9 
+32.1 
+31.0 
+21.9 
+  17.1 
+  6.5 
+  5.1 
+  0.2 

Ci.  S  

Ci.  Cu  

A.    S  

A.  Cu  

S.   Cu 

Cu 

8  ... 

Wind 

Bange  .           .             

From  Table 

42 

42 

43 

t:t 

*  Kx tract  from  Table  33,  Section  IV. 

components  of  the  distinctly  southward  and  northward  cur- 
rents in  the  different  strata.  They  are  collected  in  Table  33, 
Section  FV,  and  are  reproduced  in  Table  5. 

It  is  important  to  note  that  the  most  rapid  currents,  both 
northward  and  southward  in  the  atmosphere,  are  in  the  strato- 
cumulus  level,  2.5  kilometers  or  1.6  miles  above  the  ground, 
and  that  these  currents  decrease  in  velocity  above  and  below 
that  level.  The  eastward  velocity  averages  about  the  same  in 
the  highs  and  lows.  Hence  we  infer  that  the  strato-cumulus 
level  is  the  stratum  where  the  interchanging  motion  is  most 
rapid  between  the  Tropics  and  the  poles. 

VECTORS    OF    MOTION    IN    HIGH    AND    LOW  AREAS- — CYLINDRICAL    CO- 
ORDINATES. 

We  can  now  compute  the  true  cyclonic  and  anticyclonic  rec- 
tangular components  by  simply  subtracting  the  normal  values 
(heavy  type,  Table  3)  from  the  individual  subarea  values  in  each 
cloud  stratum.  In  this  way  the  components  «,,  r,  of  Tables  44, 
45  are  found,  and  an  example  is  given  above,  in  Table  6,  in  the 
cirrus  level  for  the  high  and  low  areas.  Against  subareas  6, 
8,  10,  12,  14,  1C,  18,  20  there  are  placed  the  corresponding 
vectors  (<r,  ft),  velocity  and  azimuth,  because  these  are  needed 
in  resolving  the  rectangular  into  cylindrical  components.  In 
the  other  subareas  the  components  are  already  in  the  N-S  and 
W-E  directions  and  they  can  be  transformed  by  mere  inspec- 
tion into  the  corresponding  cylindrical  coordinates,  which  are 
radial  and  tangential  to  a  circle  about  the  central  axis,  and  they 
give  M2,  u2,  as  in  Table  6,  and  fig.  5  following,  in  which  +  w,  = 
southward,  +  i',  =  eastward,  +  i/2  =  radial  outward,  +  r.,  = 
tangential  anticlockwise. 


HIGH. 


JIT 


LOW. 


Fio.  5.     (From  Chart  Hi. 


15 


TABLE  6.  —  Anticyclomc  and  cyclonic  components,  cirrus  9.8  kilom.* 
HIGH  AKEAS. 

Area  N«>. 

Rectangular  components. 

Cylindrical  components. 

MJ              1>,           a         fl 

"•                   »• 

1 

+  4.1       -  1.4 
+25.1      -  0.4 
+  9.2     —  6.1 
+  10.2      -  1.1 
+  16.7      -  2.6 
+  14.2     --  6.9       15.7     333 
+  1.6     +22.2 
+  2.1      -  7.2        7.5     286 
+  7.0    —  2.7 
—  15.7      -2.6       16.0     190 
+  8.0          0.0 
—  2.7     —22.5      22.8     263 
-  1.1      -  5.3 
+  6.5      -  1.7         6.8     345 
—  7.4     +8.8 
+  4.  1      -  8.3         9.2     296 
+  2.  9     —  0.  3 
—  3.  6     —  3.  5        5.  0     225 
—  2.5     —  7.5 
—  8.3      -7.8       11.4     223 

+  4.1       -   1.4 
-  0.4     —25.1 
-  9.2     +6.1 
+  1.1     +10.2 

2 

3 

4 

5 

+  16.7     .-  2.6 
+  4.9     —14.9 
+22.2     --  1.6 
-  6.6     +3.6 
-  7.0     +2.7 
+13.2      -  9.2 
0.0     +8.0 
+  14.0    —18.0 

6                          .      . 

7                        

8                          .... 

9                    

10 

11 

12 

13 

-  1.1      -  5.3 
+  3.4     --  5.9 
+  8.8     +7.4 
-  8.7    +3.0 
—  2.9     +0.3 
+  5.0          0.0 
+  7.5     —  2.5 
0.0    —11.4 

14 

15 

16 

17 

18 

19 

20 

LOW  AREAS. 

Area  No. 

Rectangular  components. 

Cylindrical  components. 

Mj                 U,              *          ft 

W*                  «, 

1                             ... 

—  16.7      +11.1 
—  2.6     +.3.4 

—16.7     +11.1 
+   3.4     +2.6 

2                               .    . 

3                       

4  

—12.6     +20.4 
—  2.6      -  5.4 
-  6.8     +6.7         9.6     136 
—10.2      -  2.3 

—20.4     —12.6 

5   

-  2.2      -  5.4 
-  0.1     +9.6 
-  2.3     +10.2 

6  

7 

8 

9 

+  1.6     +12.4 
-|-  0.6     --  9.2         9.3    .274 
+  11.7     +  7.3 
—13.1     +18.3       22.6     127 
—16.5     •-  8.7 
0.0    •-  4.3        4.3     270 
+  4.6      -  2.4 
+  1.6      -  6.2        6.4     285 

-  1.6    —12.4 

+  6.1     +7.0 
-  7.3     +11.7 
—22.5     +3.2 

10             .            .... 

11   
12  

13 

—16.5     —8.7 
-  3.0    —  3.0 
-  2.4      -  4.6 
-  5.5     +3.2 

14 

15  .. 

16  

17  

1H   

—19.2     —  3.4       19.6     190 
+  16.0     +5.1 
+  1.6     +  4.6        5.0      70 

+  16.1     —11.3 
-  5.1     +16.0 
-  2.2     +4.6 

19 

20 

TABLE  7. — Mean  components  grouped  in  three  levels.* 

MKAN   ANTK'VCI.OMC  COMPONENTS. 


•Extracts  from  Tables  44,  46  and  45,  47. 

Those  which  do  not  lie  on  the  north-south  west-east  lines 
are  transformed  as  follows:  The  coordinates  uv  v1  are  cor 
pounded  into  the  vector  (<r,  ft),  a  being  the  linear  distance 
from  the  center,  and  ft  the  angle  from  the  south.  Thus  in 
cirrus  <>,  high,  Table  44,  f/^14.2,  i']== — 6.9,  and  we  find 
<r=15.  7,  ,J=333°,  which  can  be  verified  by  reference  to  Chart 
16  a.  In  the  same  way  all  the  vectors  under  subareas  6,  8, 
10,  42,  14,  16,  13,  20  which  lie  on  the  SE-NW  and  NE-SW 
diagonals  have  been  reduced  to  vectors  (<r,  /?). 


I. 

II. 

III. 

M* 

-3.3 

+  3.9 

+  2.2 

Upper  level. 

V-L 

-4.5 

—  5.2 

-4.8 

Ci.,  Ci.  S.,  Ci.  Cu. 

a 

5.6 

6.5 

5.3 

ft 

234 

307 

294 

M,2 

0.0 

+  4.2 

-  0.8 

Middle  level. 

*l 

-  7.1 

-6.6 

-9.3 

A.  S.,  A.  Cu.,S.  Cu. 

a 

7.1 

7.9 

9.3 

13 

270 

303 

265 

«2 

+  3.3 

+  3.0 

+  1.8 

Lower  level. 

V, 

^4.1 

-  7.0 

-6.2 

Cu.,  S.,  Wind. 

a 

5.3 

7.6 

6.4 

P 

308 

294 

287 

MEAN  CYCLONIC  COMPONENTS. 


I. 

II. 

m. 

H, 

—  1.2 

—  6.8 

-  1.8 

Upper  level. 

V, 

+  10.2 

+12.3 

+  0.7 

Ci.,  Ci.  S.,  Ci.  Cu. 

a 

10.3 

14.0 

2.0 

13 

96 

119 

161 

u, 

-7.3 

+  0.3 

+  1.6 

Middle  level. 

V, 

+18.6 

+  14.4 

+  5.2 

A.  S.,  A.  Cu.,  S.  Cu. 

a 

20.0 

14.4 

5.5 

ft 

111 

89 

73 

u., 

+  0.3 

-  2.4 

-  1.5 

Lower  level. 

V, 

+  7.9 

+  6.3 

+  3.8 

Cu.,  S.,  Wind. 

a 

8.0 

6.7 

4.2 

ft 

88 

111 

112 

»  Copy  of  Table  52. 

To  reduce  these  to  the  cylindrical  coordinates,  we  subtract 
45°  from  /?  in  6  and  14,  135°  from  ft  in  8  and  16,  225°  from  ft 
in  10  and  18,  315°  from  ft  in  12  and  20.  Then  the  vector 
[a  (/?—«)]  is  resolved  at  once  into  w2,  vf  Thus  15.7,  333°  of 
Ci.  6,  high,  Table  44,  becomes  15.7,  288°;  thence  M2=4.9 
vt=  — 14.9,  as  in  cirrus  6,  high,  Table  46,  and  as  can  also  be 
verified  on  the  Chart  16  a.  In  this  way  the  coordinates  of  the 
anticyclonic  and  cyclonic  components,  Tables  46,  47,  have 
been  found.  We  have  thus  the  data  in  such  form  that  one 
more  concentration  can  be  made.  If  we  assume  that  a  sym- 
metrical gyratory  circulation  is  represented  by  the  coordinates 
of  Chart  16,  it.is  now  necessary  simply  to  take  the  mean  values 
of  the  cylindrical  coordinates  lying  on  each  circle;  that  is  to 
say,  the  mean  of  the  areas  1—4,  5-12,  13-20,  respectively. 
This  has  been  done,  and  they  are  entered  as  I,  II,  III,  in  the 
next  section  of  the  same  Tables  46,  47.  The  means  of  I,  II, 
III,  themselves,  are  also  entered  on  the  next  line,  as  average 
values  for  the  entire  circulation  in  each  cloud  level.  These 
can  be  most  conveniently  studied  by  reference  to  Charts 
17,  18,  where  specimen  vectors  are  plotted  for  the  several 
levels. 

The  rectangular  components  were  transferred  to  Chart  16 
and  drawn  so  that  the  vectors  shall  be  central  on  the  circles 
I,  II,  III,  which  run  through  the  middle  of  the  respective 


16 


adopted  subareas.  lu  order  to  get  some  idea  of  the  average 
cyclonic  nud  auticyclonic  vectors  in  the  different  levels,  the 
mean  values  of  the  vectors  found  on  the  circles  I,  II,  III,  re- 
spectively, were  taken,  and  these  give  the  relations  betvum 
the  inner  and  the  outer  portions  of  the  masses  of  air  in  motion 
in  cyclones  and  anticyclones.  They  are  shown  in  Charts  17 
and  18.  To  secure  one  more  concentration  of  the  data,  and 
to  further  eliminate  the  local  defects,  the  nine  levels  v\en> 
reduced  to  three  by  taking  the  means  of  the  three  upper,  the 
the  three  middle,  and  the  three  lower  strata  together,  respect- 
ively, and  these  are  shown  on  Chart  li).  The  occompanyiug 
small  Table  7  gives  the  corresponding  numerical  results;  it  is 
Table  52  of  the  cloud  report. 

It  is  evident,  that  it  would  be  of  great  advantage  to  meteor- 
ology to  have  similar  observations  continued  systematically  in 


the  United  States,  so  as  eventually  to  obtain  perfectly  reliable 
vectors  of  motion  throughout  the  atmosphere,  and  they  should 
be  extended  to  all  parts  of  the  world  as  rapidly  as  practicable. 
It  is  not  very  safe  to  draw  conclusions  extending  to  the  entire 
atmosphere  from  the  observations  made  at  a  few  selected  locali- 
ties, such  as  those  in  the  United  States  or  Europe,  but  it 
seems  to  be  necessary  for  us  to  do  so  in  the  present  incomplete 
state  of  meteorology.  Moreover,  we  must  use  the  material  we 
now  have  in  discussing  what  are  the  fundamental  principles  of 
dynamics  that  can  be  admitted  into  the  theory,  and  accord- 
ingly I  shall  proceed  to  take  up  the  observed  general  circula- 
tion and  the  local  circulations,  and  compare  them  with  the 
existing  theories  in  order  to  arrive  at  such  views  as  will 
probably  determine  the  theoretics  of  the  dynamic  meteorology 
of  the  future. 


III.— THE  OBSERVED  CIRCULATION  OF  THE  ATMOSPHERE  IN  THE  HIGH  AND  LOW  AREAS.1 


GENERAL     DESCKIPTION     OF     THE     VECTORS     OBTAINED     BY    OBSERVATION. 

In  my  original  report  on  the  cloud  observations  of  189G-97, 
it  was  necessary  to  present  the  data  in  such  a  form  that  other 
students  could  have  the  facts  at  first  hand.  As  then  pointed 
out  there  are  several  subareas  in  which  only  a  few  observations 
were  located,  and  they  are  quite  unevenly  distributed  about 
the  central  axis,  so  that  the  final  vectors  as  computed  do  not 
have  the  well-balanced  smoothness  which  it  is  desirable  to  ob- 
tain. The  data  was  given  in  the  form  of  tabulations  and  also 
of  diagrams,  since  it  is  easier  to  secure  from  the  latter  a  clear 
mental  picture  of  the  average  configuration  of  the  vectors  of 
motion  in  all  parts  of  the  cyclones  and  anticyclones.  Having 
done  this  at  the  outset  I  now  proceed  to  draw  up  an  average 
system  of  vectors  by  the  process  of  graphic  adjustment.  There 
will  still  remain  some  uncertainty  as  to  the  finer  details  in  cer- 
tain areas  where  the  motion  is  more  complicated,  but  I  am 
quite  sure  that  the  results  presented  in  this  paper  give  a  very 
correct  idea  of  the  mean  motions  of  the  atmosphere  over  the 
United  States  and  Canada.  It  would  require  a  good  deal  more 
labor  in  observation  and  computation  than  was  involved  in  a  '• 
single  year's  campaign  to  bring  the  work  to  that  degree  of 
perfection  which  is  desired  by  meteorologists;  this  work  must 
undoubtedly  be  expended  in  the  interest  of  science  some  time 
in  the  future.  Especially  for  the  higher  strata  of  the  high 
and  low  areas  do  we  need  more  observations,  because  the 
powerful  eastward  drift  quickly  obscures  the  comparatively 
small  gyratory  components  that  penetrate  up  to  the  high  levels. 
It  should  be  remembered  that  the  vectors  in  hand  were  pro- 
cured by  observing  the  motions  of  the  air  almost  daily  through- 
out the  year,  and  consequently  that  all  kinds  of  weather  have 
entered  our  final  results.  If  we  want  the  characteristic  circula- 
tion pertaining  to  well  developed  cyclonic  and  anticyclonic 
configurations,  it  can  be'found  only  by  selecting  the  vectors  on 
certain  days  when  these  types  are  strongly  organized,  and  dis- 
cussing them  by  themselves.  Under  the  circumstances  that 
pertained  to  the  cloud  year  we  were  obliged  to  put  every  kind 
'  of  observation  together,  without  selection,  and  this  necessarily 
produced  many  irregularities  in  the  final  scheme  of  vectors. 
I  have  now  gone  over  the  data  again,  and  by  studying  the 
balance  of  the  various  parts  of  the  system  have  brought  out 
the  revised  scheme  herewith  presented.  Its  well-balanced 
symmetry  speaks  strongly  for  its  average  accuracy,  and  it  will 
be  possible  to  draw  out  of  it  many  important  conclusions  of 
fundamental  value  for  theoretical  meteorology.  We  may  re- 
mark that  none  of  the  principles  enunciated  in  the  original 
report  have  undergone  modification  by  this  present  review. 

By  comparing  the  vectors  of  figs.  6  and  7  of  this  paper  with 
Tables  34-47  and  Charts  15  and  10  of  the  Cloud  Report,  one 
may  readily  examine  all  the  changes  that  have  been  adopted, 
and  may  also  discover  how  closely  these  charts  represent  the 
mean  system  indicated  by  the  original  observations.  Instead 
of  carrying  the  discussion  through  on  the  mean  cloud  levels 
where  the  observations  were  made,  it  is  more  convenient  to 
select  certain  planes  upon  which  the  average  vectors  are  estab- 
lished for  further  discussion. 

It  is  necessary  first  to  establish  the  normal  mean  annual 
vectors  representing  the  eastward  drift  to  which  the  observed 
vectors  are  to  be  referred,  in  order  to  decompose  them  and 
obtain  the  auticyclouic  and  the  cyclonic  vectors  by  themselves. 
These  normal  vectors  are  given  in  Table  4,  which  is  an  ex- 
tract from  Table  33,  III,  International  Cloud  Report.  The 

1  Kcprintcil  from  the -Monthly  Went  tu-r  lirvicw  for  March,  1!)02, 
F.  H.  B. 3 


eastward  velocities  are  also  represented  by  fig.  8,  Total 
eastward  velocities  in  high  and  low  areas,  which  shows  that 
the  low  areas  drift  eastward  more  rapidly  than  the  high  areas 
at  all  levels  above  the  stratus,  where  they  have  about  the 
same  velocity,  and  that  they  drift  northward  in  the  United 
States  in  the  upper  levels,  at  a  somewhat  higher  velocity  than 
in  the  low  levels.  It  is  important  to  bear  in  mind  that  the 
results  of  our  observations  pertain  only  to  the  central  portions 
of  the  North  American  Continent,  eastward  of  the  Rocky 
Mountains,  where  the  cyclonic  storm  tracks  have  on  the  aver- 
age a  northeastward  direction  toward  the  Gulf  of  St.  Lawrence. 
On  the  Rocky  Mountain  slope  they  have  a  movement  toward 
the  south  before  recurving  in  the  Mississippi  Valley.  Gen- 
erally the  eastward  drift  has  a  small  northward  or  southward 
component  varying  in  the  different  parts  of  the  world,  and  it 
is  not  quite  proper  to  draw  general  conclusions  for  the  entire 
hemisphere  from  the  motion  of  the  atmosphere  in  one  district. 
Furthermore,  since  the  cyclonic  areas  have  a  special  vortical 
progression  of  their  own,  it  seems  probable  that  the  average 
velocities  observed  in  the  high  areas  represent  the  true  motion 
of  the  total  mass  of  circulating  air  more  correctly  than  would 
the  mean  of  the  high  and  the  low  areas.  The  normal  east- 
ward and  northward  components  have,  therefore,  been  chosen 
a  little  in  excess  of  those  given  by  observation  for  the  high 
areas,  and  they  are  placed  in  Table  8. 

TABLE  8. — Normal  component  velocities  on  svx  selected  planes. 


Height. 

Eastward  ve- 
locity. 

Northward 
velocity. 

Height. 

Eastward  ve- 
locity. 

Northward 
velocity. 

Meters. 
10,000 

m.  p.  s. 

36 

m.  p.  s. 
—  2 

Miles. 
6.21 

«.&*. 

m.  p.  h. 
—  4 

7,500 

34 

-  2 

4.66 

76 

—  4 

5,000 

26 

-  1.5 

3.11 

58 

—  3 

3,000 

20 

1 

1.86 

45 

—  2 

1,000 

8 

-  1 

0.62 

17 

—  2 

Surface 

4 

—  0.5 

Surface 

9 

-  1 

Two  points  may  be  noted  in  passing:  (1)  The  eastward  drift 
seems  to  be  stratified  into  a  series  of  steps  by  a  decided  change 
of  the  eastward  velocity,  and  it  appears  that  some  form  of 
stratus  cloud  is  to  be  found  at  the  bottom,  and  some  form  of 
cumulus  cloud  at  the  top,  of  each  distinct  stratum  of  flowing 
air.  This  indicates  that  at  the  surface  of  discontinuity  between 
moving  strata,  the  stratus  type  of  cloud  forms  by  a  process  of 
cooling  through  mixture  from  adjacent  layers  of  air  at  different 
temperatures,  which  is  in  accord  with  general  theory.  It 
also  shows  that  the  cumulus  clouds  form  by  vertical  convection 
and  dynamic  cooling  within  a  stratum  having  about  the  same 
uniform  velocity  of  motion  throughout  its  mass  and  this  is 
also  theoretically  correct.  (2)  The  components  of  average 
total  motion  do  not  show  that  the  atmosphere  drifts  northward 
ill  the  higher  levels  and  at  the  surface,  but  southward  in  the 
lower  middle  levels,  somewhat  elevated  from  the  ground,  as 
was  claimed  should  be  the  case  by  Professor  Ferrel  in  his  canal 
theory  of  the  general  circulation  of  the  atmosphere.  I  will 
return  to  this  topic  and  consider  it  at  length,  but  the  fact  here 
indicated  is  that  the  observations  do  not  sustain  that  part  of 
the  general  canal  theory.  It  is  becoming  clearly  demonstrated 
to  students  that  the  circulation  of  the  air  is  a  more  complicated 
problem  than  the  early  meteorologists  assumed,  and  in  conse- 
quence it  will  be  necessary  to  study  in  detail  the  stream  lines 
over  the  several  continents  and  oceans,  find  out  their  local 

17 


20 


characteristics,  and  after  that  try  to  combine  them  in  a  large 
comprehensive  scheme. 

DESCRIPTION    OF    THE    CIRCULATION    OVEK    HIGH    AND  LOW  AREAS. 

Figs.  6  and  7  represent  the  adjusted  mean  vectors  of  direc- 
tion and  velocity  of  motion  in  high  and  low  areas,  as  derived 
from  the  Weather  Bureau  observations  of  1896-97.  They  are 
based  upon  about  6,000  theodolite  observations  made  at 


Flo.  8. — Total  eastward  velocities  in  high  and  low  areas. 

Washington,  D.  C.,  and  about  25,000  nephoscope  observations 
made  at  15  stations  distributed  quite  uniformly  over  the  terri- 
tory east  of  the  Rocky  Mountains.  They  give  only  a  mean  or 
average  scheme  of  the  circulation  and  are  necessarily  somewhat 
idealized,  as  regards  the  movements  of  the  air  in  individual 
configurations,  since  they  include  all  the  anticyclones  and 
cyclones  of  the  cloud  year,  many  of  which  were  only  imper- 
fectly developed,  and  could  not  have  agreed  with  the  best 
types  that  might  have  been  selected.  In  order  that  no  false 
impressions  should  remain  with  students  concerning  the  actual 
circulation  of  the  atmosphere,  because  of  this  construction  of 
a  well-balanced  type,  I  compiled  for  the  International  Cloud 
Report  a  series  of  composite  charts,  Nos.  20  to  35,  inclusive, 
which  show  the  actual  stream  lines  in  high  and  low  areas  over 
the  several  areas  of  the  United  States,  both  for  summer  and 
winter.  These  charts  are  not  only  interesting,  but  they  are 
very  valuable,  because  they  give  the  normal  flow  of  the  air 
when  the  anticyclonic  and  cyclonic  centers  are  located  in  dif- 
ferent parts  of  the  country.  They  ought  to  be  studied  care- 
fully by  every  forecaster,  and  the  general  knowledge  given  by 
the  charts  should  be  kept  firmly  in  mind  when  considering 
the  meaning  of  the  individual  daily  weather  maps,  as  they  will 
guide  the  judgment  to  safer  conclusions  than  would  be  possi- 


ble without  them.  For  the  student  of  theoretical  meteorology 
they  are  indispensable,  because  they  correct  the  impressions 
which  may  be  given  by  a  contemplation  of  the  figs,  (i  and  7,  or 
by  reflecting  upon  the  analytical  formula). 

DISCUSSION    OF    THE    VECTORS    IN    HIGH    AREAS. 

The  area  about  the  center  of  circulation  was  subdivided  into 
twenty  small  parts,  numbered  as  already  described  in  a  previous 
paper;  the  upper  left-hand  plans  of  figs.  6  and  7  show  them 
again  for  convenience  of  reference.  Through  the  center  of 
each  of  the  three  concentric  groups  a  circle  is  drawn  in  dotted 
lines,  and  these  are  marked  I,  II,  III,  their  distance  from  the 
center  being  250,  750,  1,250  kilometers,  respectively.  The 
adopted  heights  of  the  planes  of  motion  in  meters  and  miles 
are  written  on  each  level,  also  the  normal  velocity  vector  in 
meters  per  second  (in.  p.  s. ),  and  miles  per  hour  (in.  p.  h. ).  The 
scale  of  distances  is  1  cm.  =  500  kilometers,  and  the  scale  of 
velocities  is  1  mm.  =  2  meters  per  second;  the  latter  can  be 
reduced  to  miles  per  hour  by  multiplying  with  the  factor  2.24. 
The  left-hand  plans  contain  the  total  vector  as  observed  in  the 
atmosphere;  the  right-hand  plans  give  the  component  vector, 
which,  combined  with  the  normal  vector,  produces  the  observed 
vector,  using  the  rule  of  the  parallelogram  of  vectors.  Each 
vector  has  been  carefully  constructed  and  deserves  consider- 
able confidence.  The  smoothly  balanced  configuration  in  each 
level  and  the  gradual  change  which  occurs  in  passing  from 
one  level  to  another  show  that  this  represents  a  natural  and 
easy  form  of  flow  for  the  atmosphere,  so  that  the  motion  will 
occur  without  sharp  changes.  The  figures  speak  plainly  for 
themselves,  and  only  a  few  words  are  required  regarding  the 
distinguishing  features.  In  the  high  areas  the  total  flow  di- 
minishes in  strength  from  10,000  meters  to  the  surface;  it  has  a 
slight  curvature  northward  over  the  center  in  the  highest  level, 
but  this  concavity  of  the  curves  gradually  increases  till  in 
the  lower  levels  and  at  the  surface  the  sinuous  lines  are  con- 
verted into  auticyclonic  gyrations.  The  vectors  north  of  the 
center  are  longer  than  those  south  of  it  from  the  top  to  the 
bottom.  There  is,  however,  a  strong  eastward  drift  in  all 
levels,  inward  on  the  west  side  and  outward  on  the  east  side, 
which  is  -never  overcome. 

Passing  now  to  the  anticyclonic  component  vectors,  it  is 
noted  that  there  is  a  remarkable  symmetry  in  the  configura- 
tion from  the  highest  level  to  the  lowest,  taken  as  a  whole. 
There  are,  however,  two  special  features  to  be  observed:  (1)  In 
the  central  areas,  I,  the  flow  is  inward  on  the  highest  level, 
more  from  the  north,  however,  than  from  the  south;  it  is  tan- 
gential on  the  middle  level;  and  it  is  outward  in  the  lowest 
level.  This  indicates  a  type  of  true  vortex  motion,  which  pre- 
vails at  the  center  of  anticyclones,  and  by  it  the  air  is  drawn 
in  at  the  top  and  discharged  at  the  bottom  of  the  vortex  tube. 
(2)  On  the  middle  areas,  II,  the  flow  is  nearly  tangential 
throughout  the  entire  series  of  strata,  but  on  the  outer  areas, 
III,  the  vectors  are  pointed  slightly  outward  from  the  top  to 
the  bottom,  though  more  strongly  on  the  east  side  than  on  the 
west  side.  There  is,  furthermore,  the  special  feature  that  at 
the  south  or  southwest  side  of  the  anticyclouic  area,  near  the 
place  marked  A,  a  distinct  discontinuity  occurs  in  the  vectors, 
bv  which  on  the  west  side  an  inflow  from  the  south  takes  place, 
and  on  the  east  side  an  outflow  from  the  north  is  indicated, 
interpret  these  two  facts  together  to  mean  that  in  the  south- 
east quadrant  there  is  a  tendency  for  a  heavy  stream  of  the 
general  circulation  from  the  northwest  to  divide,  so  that  a 
large  portion  moves  to  the  south  side  of  the  adjacent  cyclonic 
area  and  a  small  portion  curls  westward  about  the  center  of 
the  high  area.  Also,  on  the  west  side  of  the  high  area  a  stream 
from  the  south  divides,  part  flowing  over  the  north  of  the  high 
area  and  another  part  curling  about  the  north  side  of  the  cen- 
ter of  the  adjacent  low  area.  Fig.  9,  Curling  of  the  northward 
uid  southward  streams  about  the  centers  of  high  and  low  areas, 


21 


FIG.  9.— Curling  of  the  northward  and  southward  streams  about  the  centers  of  high  and  low  areas. 


gives  an  idea  of  this  process,  especially  in  the  strato-cumulus 
level,  or  at  about  3,000  meters  elevation.  The  heavy  broken 
line  represents  the  resulting  sinuous  eastward  flow  at  that 
level.  In  the  flow  of  fluids  a  wave  motion,  when  the  velocity 
exceeds  a  given  amount,  collapses  and  reappears  in  the  form 
of  whirls  of  discontinuous  surfaces  along  the  sides.  Some- 
thing of  this  sort  is  apparently  operating  in  this  connection. 
We  observe  that  in  the  3, 000-meter  level  the  anticyclonic 
vectors  are  stronger  than  in  the  levels  above  or  below,  the 
diminution  toward  the  surface  being  greater  than  toward  the 
higher  levels.  The  superposition  of  the  component  gyration 
upon  the  eastward  drift  is  distinct  and  even  vigorous  at  10, 000 
meters,  and  hence  it  is  inferred  that  the  disturbance  of  the 
atmosphere  in  high  areas  extends  to  at  least  6  or  8  miles, 
though  only  as  a  small  deflection  of  the  eastward  drift  in  the 
upper  strata. 

TABLE  9. — Rectangular  and  cylindrical  coordinates  in  high  areas. 


DISCUSSION    OF    VECTORS    IN    LOW    AREAS. 

The  vectors  in  the  low  areas  should  in  general  be  a  little 
longer  than  those  in  the  high  areas.  In  nature  the  highs  cover 
a  larger  territory  than  do  the  lows,  but  as  the  amount  of  air 
which  streams  through  each  of  them  is  probably  about  the 
same,  it  would  require  a  greater  velocity  in  the  lows  to  pro- 
duce an  equal  discharge  through  them.  The  vectors  flow 
southward  relatively  to  the  center,  and  they  are  larger  on  the 
southern  side  than  on  the  northern.  The  connection  of  the 
streams  between  the  high  and  low  areas  is  shown  by  the  smooth 
flow  of  the  two  sets  of  vectors  on  their  eastern  and  western 
sides,  respectively.  The  stream  lines  are  convex  upward,  and 
the  curvature  increases  from  the  10,000-meter  level  to  the  sur- 
face. In  the  1, 000-meter  level  the  gyratory  movement  nearly 
supersedes  the  sinuous  or  wave-like  flow,  but  the  vectors  on 
the  north  side  are  not  entirely  reversed  to  the  westward. 
TABLE  10. — Rectangular  and  cylindrical  coordinates  in  low  areas. 


•jj 

-  t; 

r  £ 

h 

S  X 

i- 

10,  000  meters. 

7,  500  meters. 

82 

a  o 

1s 

|l 

a  £ 
^  ji 

10,  000  meters. 

7,  500  meters. 

ll 
.38 

|g 

-  = 

^  s 
a 

«,           », 

«,         «, 

«,           «, 

«,         «, 

*l 

a 

u,           vl 

M.2              V.t 

tt,           1), 

M.J              1)2 

s| 

S 

1 

—  2     +29 

—  2     —  6 

+  2     +'24 

+   2     —10 

s 

1 

—10     +46 

—10     +10 

—  6     +46 

—  6     +12 

T 

E 

2 

+  6     +31 

-  4     •-  6 

+  8     +30 

—  4    —  8 

i. 

25O 

E 

2 

—12     +34 

—  2     +12 

—12     +32 

—  2     +12 

250 

N 

3 

+  6     +40 

—  6     —  5 

+  6     +36 

—  6     —  2 

km. 

N 

3 

—  4     +32 

+  4+4 

—  6     +26 

f  6     +  8 

km. 

W 

4 

+  4     +39 

-  3     +5 

—  4     +32 

+  2—4 

W 

4 

—  6     +40 

—  6     —  4 

+  4     +44 

-10     +  4 

S 

5 

1                       j    "** 

+  8     +28 

•            1 

+  8—8 

0     +24 

0    —10 

S 

5 

—  8     +44 

—  8+8 

—10     +44 

—10     +10 

SE 

6 

+  10     +30 

+  2—9 

+  8     +30 

+  3—9 

SE 

6 

—12     +42 

—  5     +14 

—12     +34 

—  6     +10 

E 

7 

+  7     +40 

+  4-7 

+  8     +36 

+  2—8 

E 

7 

—10     +32 

—  4     +10 

—10     +30 

—  4     +10 

NE 

8 

+  6     +43 

+  2—7 

+  8     +38 

—  4    —  8 

II. 

750 

NE 

8 

—  4     +24 

—  4     +12 

—  6     +26 

-4+9 

• 
750 

N  " 

9 

+  2     +44 

+  2—8 

+  4     +40 

—  4    —  6 

km. 

N 

9 

—  6     +32 

+  4+6 

—  4     +28 

+  4+6 

km. 

NW 

10 

—  4     +40 

—  2     —  6 

—  8     +38 

+  4    .-8 

NW 

10 

—  3     +30 

+  6+3 

+  4    +26 

+  4+8 

W 

11 

—  4     +36 

—  1     —  4 

—10     +30 

+  4     —10 

W 

11 

+  6     +44 

—  6+8 

+  8     +42 

—  8+8 

SW 

12 

—  5     +28 

0—7 

—  8     +30 

—  4     —  8 

SW 

12 

+  2     +46 

—  6+8 

0    +46 

—  7     +10 

^  B^— 

s 

13 

+  8     +30 

+  8—6 

+  4     +24 

+  4     —10 

S 

13 

—  4     +.44 

-4+8 

—  4     +42 

-  4     -f    8 

SE 

14 

+  9     +28 

+  2—9 

+  6     +28 

0—9 

SE 

14 

—  8     +40 

-4+8 

—  8.   +38. 

-4+8 

E 

15 

+  9     +40 

+  4-9 

+  10     +38 

+  4     —10 

1 

E 

15 

—  6     +36 

0+6 

—  8     +30 

-4+8 

NE 

16 

+  10     +42 

—  4    —10 

+   9     +40 

—  4—8 

III. 

1O  f*("l 

NE 

16 

—  4     +28 

—  4+8 

—  8     +28 

+  2     +10 

III. 

1,250 

N 

17 

-  2     +44 

+  2     —  8 

+  3     +42 

—  3     —  8 

,  i'.Ml 

km. 

N 

17 

—  4     +30 

+  •4     +  6 

—  4     +30 

+  4+4 

km. 

NW 

18 

-  8     +40 

+  4    .-8 

-  6     +40 

+  1-8 

NW 

18 

—  4     +30 

+  6+4 

-  2     +28 

+  6+2 

W 

19 

—  8     +32 

+  4-8 

—  9     +30 

+  4-9 

W 

19 

+  6     +42 

—  6+6 

+   6     +40 

—  6+6 

SW 

20 

—  8     +32 

-4—6 

—  8     +28 

-6—8 

SW 

20 

+  4     +42 

-4+6 

+  4     +40 

—  2+7. 

22 


TMH.K  9.  —  Rectangular  and  cylindrical  coordinate*  in  high  areas  —  Cont'd. 

TABI.E  10.  —  Rectangular  and  cylindrical  coordinate*  in  low  arm*     fmit'il. 

nS 

S,  000  meters. 

3,  000  meters. 

«£ 

ll 

S,  000  meters. 

3,000  meters. 

$3 

2  ~~~ 

1? 

SJS 
«I 

11 

•—  S 

11 

5  • 

|| 

1; 

sl 

Q 

u,          t), 

u,         v, 

«,          «1 

u,         «, 

°J 

5I 

n 

ul            vl 

«2               U, 

U,             «, 

«!               «i 

y 

S 

1 

+  2     +18 

+  2-8 

+  4     +12 

+  4-8 

s 

1 

—10     +42 

—  10     +16 

—10     +40 

—10     +20 

E 

2 

+  8     +22 

—  2    —  8 

+  8     +16 

-4—8 

I. 

250 

E 

2 

—16     +22 

-  4     +16 

—18     +16 

-  4     +18 

i. 

250 

N 

3 

+  4+34 

—  4     —  8 

—  2    +26 

+  2—6 

km. 

N 

3 

-  2     +16 

+  2     +10 

-  8     +12 

+  8+6 

km. 

W 

4 

—  8    +28 

—  2     —  8 

—  8     +16 

+  4-8 



W 

4 

+  14     +28 

-  6     +14 

+  16   .+28 

-  8     +16 

,s 

5 

+  4     +16 

+  4    —10 

+  6     +14 

+  6    .—  .6 

S 

5 

-  4     +36 

-  4     +10 

—1C)    +32 

—1(1     +12 

SE 

6' 

+  8    +20 

+  4-8 

+  12     +14 

+  4    —12 

SE 

G 

—14     +32 

-  5     +14 

—12    +26 

-  G     +12 

E 

7 

+  8    +28 

+  2—8 

+10    +20 

0    —10 

E 

7 

—12     +24 

-  2     +12 

—14     +24 

—  4     +14 

NE 

8 

+  G     +32 

0—8 

+  6     +28 

+  2     —10 

II. 

750 

NE 

8 

—10     +20 

+  4     +12 

—12     +12 

+  4     +14 

II. 
750 

N 

9 

+  3    +34 

—  3     —  8 

—  2     +28 

+  2—8 

km. 

N 

9 

—10     +18 

+  6     +10 

—12     +14 

+  6     +12 

km. 

NW 

10 

—  6     +30 

+  1    -  7 

—  8    +24 

+  2    —10 

NW 

10 

+12    +20 

-  6     +12 

+  12     +18 

—  7     +10 

W 

11 

—  8    +24 

+  2—8 

—  8     +24 

-  4    —  8 

W 

11 

+12    +30 

-  4     +12 

+  16     +22 

-  2     +16 

SW 

12 

—  6     +20 

0—8 

-  6     +14 

—  4    —  8 

SW 

12 

+  4     +38 

-  4     +12 

+  12     +32 

0     +18 

;S 

13 

+  4     +16 

+  4    —10 

+  8     +10 

+  8     —10 

s 

13 

-  4     +32 

-4+6 

—  G     +28 

—  G     +   8 

SE 

14 

+  8     +22 

+  4-8 

+10     +16 

+  5    —11 

SE 

14 

—  G     +30 

-  4     +6 

—  8     +26 

—  2     +10 

E 

15 

+  8     +30 

+  4-8 

+  10     +26 

+  6     —10 

E 

15 

—  8    +22 

-4+8 

—10     +18 

-  2     +10 

NE 

16 

+  7     +32 

0    —10 

+  10     +28 

—  2     —12 

III. 

1,250 

NE 

16 

-  6     +22 

+  2     +7 

—10     +14 

+  4     +10 

III. 

1,250 

N 

17 

0    +36 

0    —10 

+  2     +30 

—  2     —10 

km. 

N 

17 

—  8     +20 

+  6+8 

—12     +14 

+  12     +   6 

km. 

NW 

18 

—  9     +32 

+  1  -11 

—12     +24 

+  4    —12 

NW 

18 

+  6     +22 

—  2+7 

+  10     +18 

-6+8 

W 

19 

—10    +24 

+  2    —10 

—10    +24 

—  4    —10 

W 

19 

+  8     +28 

-2+8 

+  12     +24 

—  4     +12 

SW 

20 

—  8     +22 

-  4     —  8 

-  8     +14 

—  4     —10 

SW 

20 

+  4     +34 

-4+8 

+  4     +28 

-4+8 

§1 

1,000  meters. 

Surface. 

%S 

ti 

c  i 

h 

1,000  IIIC'tCTS. 

Surface. 

gl 

1! 

ii 

a  s 
1* 

p 

ll 

=  ? 

Sj 

a 

«1          «1 

«,            D, 

«,           1>, 

«.         *• 

5I 

sl 

a 

U,              1), 

U,            V, 

'   w,         r, 

«.         «t 

si 

S 

1 

+  4+2 

+  4-6 

+   3             0 

+  3-4 

s 

1 

—  6     +24 

-  G     +1G 

—  4     +10 

-4+6 

E 

2 

+  6     +12 

+  4—6 

+  3+7 

+  3—3 

I. 

250 

E 

2 

-8+4 

-  4     +  8 

-  6          0 

-4+6 

I. 

'27(1 

N 

3 

-  6     +14 

+  6—6 

-4+7 

+  4-3 

km. 

N 

3 

+  10     +   4 

^10     +  4 

+  4    .-  2 

—  4+6 

km. 

W 

4 

—  8+6 

+  2—8 

-5+2 

+  2-5 

W 

4 

+  10     +12 

-  4     +10 

+  8+8 

-4+8 

S 

5 

+  8+4 

+  8-4 

+  3+1 

+  3—3 

S 

5 

-  4     +20 

-  4     +12 

-  4     +10 

-4+5 

SE 

6 

+10    +  6 

+  6    —  8 

+  4+2 

+  4-4 

SE 

6 

—10     +12 

-  4     +10 

—   6+6 

—  2+6 

E 

7 

+  8     +10 

+  2—8 

+  «     +  8 

+  4-6 

E 

7 

—10     +  6 

-  2     +10 

—  6+2 

—  2+6 

NE 

8 

+  6     +16 

+  6—8 

+  4     +10 

+  2—7 

II. 

750 

NE 

8 

—10     +  8 

+  8+6 

-  4    —  2 

-2+4 

II. 
760 

N 

9 

—  4     +16 

+  4-8 

-  3     +10 

+  3—6 

km. 

N 

9 

+  10     +  4 

—10     +  4 

+  4-2 

—  G     +4 

km. 

NW 

10 

-  8     +10 

+  1-9 

—  5+8 

0    —  6 

NW 

10 

+  12     +   4 

-  6     +10 

+  6+2 

-4+5 

W 

11 

—10    +  8 

0    —10 

—  G     +  2 

+  2—6 

W 

11 

+  8     +14 

-6+8 

+  6+8 

-4+6 

SW 

12 

—  8+2 

-  2     —10 

—  3           0 

+  2-5 

SW 

12 

+  8     +20 

-  4     +14 

+  4     +10 

—  2+8 

S 

13 

+  8          0 

+  8—8 

+  4-1 

+  4-5 

S 

13 

-  4     +18 

-  4     +10 

-4+8 

-4+4 

SE 

14 

+  10     +  4 

+  6—9 

+  7+5 

|    6    —  4 

SE 

14 

—10     +10 

-  2     +10 

-6+4 

-4+4 

E 

15 

+  8     +14 

+  6—8 

+  6+9 

+  5—6 

E 

15 

-8+8 

0+8 

-6+4 

0     +  6 

NE 

16 

+  8     +16 

+  2    —11 

+  4     +10 

+  2—7 

III. 

1,  250 

NE 

16 

-  8     +10 

+  8+4 

—  4     —  2 

-2+4 

III. 

1,250 

N 

17 

-  4     +18 

+  4    —10 

-4+8 

4-  4     —  4 

km. 

N 

17 

+  10     +  6 

—10    +  2 

+  4-2 

—  6+4 

km. 

NW 

18 

—10    +12 

+  4    —10 

—  6+8 

+  1-7 

NW 

18 

+  10     +  8 

-9+4 

+  6          0 

—  4     +6 

W 

19 

—10    +10 

-  2    —10 

-7+4 

0—7 

W 

19 

+  8     +14 

—  6+8 

+  6+8 

—  4+6 

SW 

20 

—10    +  4 

—  4     —10 

—  5    —  1 

—  2    —  5 

SW 

20 

+  6     +14 

0+8 

+  3+8 

—  2+5 

"i      southward. 
PJ  =  eastward. 


S  =  radial  outward. 

2  =  tangential  counter  clockwise. 


4-  MI  =  southward. 
+  i'i  —  eastward. 


y=  radial  outwiinl. 

•»  =  taiiK<>!iti!i]  counter dockfrtae. 


The  cyclonic  components  are  very  symmetrically  formed 
throughout  the  entire  stratum  of  air  that  has  been  examined. 
They  have  the  following  characteristic,  namely,  that  from  the 
surface  to  the  10,000-meter  level  the  vectors  have  an  inflow 
toward  the  center,  except  in  a  few  subareas  marked  with  the 
letter  A.  It  is  noted  that  from  the  10,000-meter  level  to  the 
l,00p-meter  level,  near  the  place  A,  the  vectors  are  almost  ex- 
actly opposed  to  each  other  in  direction,  those  on  the  east  side 
flowing  outward  and  those  on  the  west  side  flowing  inward. 
This  divergence  of  direction  indicates  that  a  stream  flows  from 


the  north  to  the  south  on  the  west  of  the  low  area,  and  that 
an  independent  stream  flows  northward  on  the  east  side  of  the 
]pw  area,  something  in  the  manner  suggested  on  fig.  9.  The 
separate  streams  from  the  north  and  from  the  south  coalesce 
on  the  south  side  of  the  center  of  the  low  area,  us  they  do  on 
the  north  side  of  the  high  area,  but  the  two  streams  have  an 
origin  milnidi-  the  areas  of  high  and  low  pressure,  respectively. 
Furthermore,  it  is  noted  that  while  in  the  high  area  the  posi- 
tion of  the  point  A  is  nearly  stationary  in  all  the  strata  mapped 
out,  on  the  contrary  it  rotates  nearly  00°  from  the  east  of 


23 


north  at  the  surface  to  the  north  of  west  in  the  highest  stratum. 
The  stream  of  warm  air  from  the  south  curls  around  toward 
the  west  as  it  ascends  from  the  surface  to  the  upper  levels, 
making  a  quarter  of  a  helical  revolution  in  an  ascending 
spiral.  The  length  of  the  vectors  is  greatest  in  the  3, 000- 
meter  level,  2  miles  above  the  ground,  and  the  vectors  become 
gradually  shorter  upward  and  downward,  diminishing  more 
rapidly  toward  the  surface.  This  agrees  with  the  system  of 
vectors  in  high  areas,  and  shows  that  the  influence  of  the 
cyclone  extends  above  the  10, 000-meter  level,  where  it  still 
deflects  considerably  the  eastward  drift,  though  it  is  most 
vigorous  in  the  3,000-meter  level.  The  length  of  the  vectors 
increases  gradually  from  the  Ill-areas  to  the  I-areas,  and  aver- 
ages about  twice  as  long  in  the  latter  as  in  the  former.  In  the 
auticyclonic  components  the  Ill-vectors  are  even  longer  than 
the  I-vectors,  and  they  do  not  have  any  agreement  with  the 
simple  vortex  law  m  <«  =  constant,  where  a  is  the  radial  distance 
from  the  axis  of  rotation,  and  <u  the  angular  velocity. 

In  the  cyclonic  components  the  I-vectors  are  longer  than 
the  Ill-vectors,  but  they  fall  short  of  exact  conformity  with 
the  pure  vortex  theory.  The  entire  flow  suggests,  therefore, 
the  conflict  of  two  couuterflowing,  horizontal  streams  which 
tend  to  produce  vortical  rotation,  but  in  fact  fail  to  reach  this 
ideal,  except  possibly  in  highly  developed  cases  of  severe 
storms.  There  is  no  evidence  that  these  motions  are  primarily 
due  to  vertical  convective  currents  developed  through  the  local 
heating  or  cooling  of  restricted  areas  near  the  center  of  the  cy- 
clonic and  anticyclonic  areas,  respectively.  It  is  evidently  de- 
sirable to  avoid  extreme  statements  in  this  connection,  because 
a  study  of  the  motions  of  the  atmosphere  shows  that  nearly 
every  possible  type  of  motion  from  the  counterflow  of  opposing 
horizontal  streams  to  the  pure  vortex  due  to  an  ascending  helix 
may  occur,  and  yet  the  present  compilation  indicates  that  the 
former  is  the  average  type  to  which  the  stream  lines  conform 
in  the  extra-tropical  circulation  of  the  United  States. 

THE    NUMERICAL    VALUES    OF    THE    VECTORS. 

In  order  to,  bring  out  these  facts  a  little  more  clearly,  the 
vectors  of  tig.  G  have  been  translated  into  the  numerical  values 
of  Table  9,  Rectangular  and  cylindrical  coordinates  in  high 
areas;  and  those  of  fig.  7  into  the  numbers  of  Table  10,  Rectan- 
gular and  cylindrical  coordinates  into  low  areas.  These  tables 
need  no  further  explanation  in  this  connection,  after  what  has 
been  already  stated. 

Table  11,  Mean  components  on  the  I,  II,  III  circles  in  meters 
per  second  and  in  miles  per  hour,  is  derived  from  the  anti- 
cyclonic  components  of  Table  9,  and  the  cyclonic  components 
of  Table  10,  by  taking  the  arithmetical  mean  of  the  I-areas 
(1-4),  the  Il-areas  (5-12),  and  the  Ill-areas  (13-20).  These 
means  give  the  average  value  of  the  motion,  though  we,  of 
course,  depart  from  the  perfectly  natural  condition  by  the 
summation.  Thus  in  the  anticyclonic  areas  for  the  radial  com- 
ponent 11. ^  there  is  an  inflow  at  the  top  of  I-areas,  and  an  outflow 
at  the  bottom;  and  a  gentle  outflow  in  the  Il-areas  and  Ill- 
areas  from  the  top  to  the  bottom.  Also  compare  fig.  10,  where 
the  results  of  Table  11  are  plotted.  The  tangential  com- 
ponent r,2  is  stronger  throughout  the  middle  strata  than  in 
those  which  are  higher  or  lower,  bvit  it  is  much  more  vigorous 
in  the  Ill-areas  than  in  the  I-areas  especially  at  the  3, 000- 
meter  level.  In  the  cyclonic  areas  the  radial  component  «2  in- 
creases generally  from  the  Ill-area  to  the  I-area.  There  is  a 
little  irregularity  in  the  changes  of  this  component  probably 
due  to  imperfections  in  my  vector  system.  The  tangential 
component  i',  increases  rapidly  from  the  Ill-areas  to  the  I-areas, 
and  remarkably  so  at  the  3,000-meter  level. 

It  has  been  taught  in  the  common  expositions  of  the  canal 
theory  of  the  general  circulation  that  there  exists  in  middle 
latitudes  a  strong  northward  component  in  the  upper  strata, 
a  strong  southward  component  in  the  surface  and  lower  strata, 


and  a  powerful  eastward  component  in  all  strata,  increasing 
irom  the  ground  upward.     It  can  be  seen  by  inspecting  figs. 

TABLE  11. — Mean  components  on  I,  II,  III  circles. 
ANTICYCLONIC  COMPONENTS. 


Oistanci-  from 
center. 

I. 

250  kilometers. 

750k 

ii. 
lometers. 

in. 

1,250  kilometers. 

Meters  per  second. 

U.                        V} 

~ 

V 

H=10,000 
7,500 
5,000 
3,000 
1,000 
0 

—  3.8     —  3.0 

—  1.5    ••-  6.0 
—  1.5      -  8.0 
+  1.5     —  7.5 
+  4.0    —  6.5 
+  3.0    —  3.8 

+  1.9      -  7.0 
+  0.1    —8.4 
+  1.3      -  8.1 
+  1.0     —  9.0 
+  3.1      -  8.1 
+  2.5      -  5.4 

+  2.0     —  8.0 
0.0    —  8.8 
+  1.4    --  9.4 
+  1.4     —10.6 
+  3.0    —  9.5 
+  2.5     —  5.6 

CYCLONIC  COMPONENTS. 

H—  10,  000 

7,500 
5,000 
3,000 
1,000 
0 

-  3.5     +5.5 
-  3.0    +9.0 
-  4.5     +14.0 
—  3.5     +15.0 
—  6.0     +9.5 
—  4.0     +6.5 

-  2.9     +8.6 
—  3.9     +8.9 
—  1.9     +11.8 
—  2.4     +13.5 
—  3.5    +9.3 
—  3.3     +5.5 

—  1.5     +  6.:5 
—  1.0    +6.6 
-1.5     +7.3 
-  1.0     +9.0 
-2.9     +6.8 
-  3.3     +4.9 

ANTICYCLONIC  COMPONENTS. 

Distance  from 
center. 

I. 
155  miles. 

II. 
466  miles. 

III. 

777  miles.      > 

Miles  per  hour. 

"s                "s 

"2                      *J 

"2                vi 

H—  10,  000 

7,500 
5,000 
3,000 
1,000 
0 

-  8.5    —  8.7 
_  3.4     —13.4 
—  3.4     —17.9 
+  3.4     —16.8 
+  8.9     —14.5 
+  6.7     --  8.5 

+  0. 
+  2. 
+  2. 
+  6. 
+  5. 

3    —15.  7 
2    —18.8 
9     —18.  1 
2     —20.  1 
9     —18.1 
6     —12.1 

+  4.  5    —17.  9 
0.0    —19.7 
+  3.1     —21.0 
+  3.1     —23.7 
+  6.7     —21.3 
+  5.6     —12.5 

CYCLONIC  COMPONENTS. 

H—  10,  000 
7,500 
5,000 
3,000 
1,000 
0 

—  7.8     +12.3 
—  6.7     +20.  1 
—  10.1     +31.3 
—  7.8     +33.6 
—13.4     +32.4 
-  8.9     +14.5 

—  6. 
—  8. 
-  4. 
—  5. 
-  7. 
-  7. 

5     +19.2 
7     +19.9 
3     +26.4 
4     +30.  2 
8     +20.  8 
4    +12.3 

—  3.4     +14.5 
-  2.2     +14.8 
-  3.4     +16.3 
—  2.2     +20.1 
—  6.5     +15.2 
—  7.4    +11.0 

TABLE  12.—  Northward  and  southward  velocities  in  selected  areas. 

Hoi 
oft 

Northward. 

Southward. 

S"t           L.  16,  8,  2,  7,  15.  6.  14. 

H.  16,  8,  2,  7,  15,  6,  14. 
L.  18,  10,4,  11,  19,  12,20. 

stratum. 

M,                  -I), 

1 

10,  000            -  6.  4      +34.  5 
7,500            -  8.4       +31.9 
5,000           —  9.1       +25.2 
3,000           —10.3       +19.7 
1,000            -  9.2       +7.9 
Surface          -  5.2      +2.6 

+  4.4      +37.7 
+  5.8       +36'.  2 
+  8.1       +27'.  6 
+  10.6      +22.7 
+  8.4      +111  7 
+  5.3      +  6l9 

Compare  Table  124,  International  Cloud  Report,  p.  606. 

Anf/cyc/o/j/c  Area. 

f—\       /  *    / 

ncta/a/  component  t/2-  -"z-''r>wc/,--t-u2oufiva'/-£/ve/oc/fy: 

/0000m 
7600 

6OOO 
3000 
/OOO 

o 

—2        0-1-2                         —2         0     -f-2                          -20     +2 

\ 

\ 

( 

/ 

/ 

' 

l 

\ 

\ 

\ 

X 

^^^ 

! 

^ 

N 

\ 

S 

\ 

N 

L 

ff 

/ 

//) 

r) 

Anf/cyc/on/c   Area. 
Tangent/a/  component  V2-    ~^z=  c/ocA-w'se  rofcrf/or?. 

'0000m. 
7SOO 

5000 
3000 

/OOO 
O 

-S     —ff     -^    —2        O     -f-2          -<S     —&     —4     -2        O     +2           -/O     -<S     —6     -4-      —2        O 

^ 

/ 

/ 

/ 

7 

I 

I 

7 

I 

[ 

t 

/ 

P 

, 

\ 

\ 

X 

^> 

**** 

*^> 

J 

v. 

•^.. 

-^~ 

ff 

^^ 

— 

f«^. 

—— 

^ 

^ 

Cyc/ofi/'c  Area  . 

/7aof/a/  component  u2-    ~  U2  -  ''w&>^ve/oc/Yy. 

750O 

5000 
3000 

/OOO 
O 

-  2        0     -f-2                          —2O-+-2                          —2         0     -f-2 

/ 

) 

v 

^ 

: 

^ 

•^ 

\ 

) 

f 

**~ 

^- 

/ 

, 

/ 

/ 

/ 

^ 

r 

\ 

ff 

I 

1 

Cyc/on/c  Area. 

•    •••                                         /               y 

/anqenf/a/  component    V2-  +^2-  «»^ofocfrwfse^  rrtarffan^ 

/OOOOrn. 
7600 

SOOO 
3OOO 

/OOO 
0 

0    +2     -t-4     +6    +8    +/O  -+/2    -t-/4  O     +2     -+±4    -t-ff    -hd    -+•/(?    +/2        0    -t-2    -+4    -+6     -hff 

S 

S 

s 

^^ 

\ 

x 

^. 

V. 

v 

s 

\ 

S 

^ 

1 

\ 

\ 

s 

•^ 

\ 

\ 

r 

.-•• 

^ 

^* 

^— 

^^ 

»-* 

X 

--- 

r^ 

*^ 

*s 

' 

.s 

/ 

/ 

j 

7i 

r- 

*-* 

«*• 

^** 

Tt 

j 

/ 

^ 

Flo.  10. — Radial  and  tangential  components  in  anticyclonic  and  cyclonic  areas.     From  Table  11. 


25 

• 

6  and  7  that  while  there  is  everywhere  a  general  eastward  <  of  the  low  there  is  a  southward  current  also  strongest  in  the 
drift,  there  are  certain  subareas  over  which  especially  a  north-  j  same  level.  The  interchange  of  air  between  the  pole  and  the 
ward  component  prevails,  and  others  over  which  there  is  a  Tropics  appears,  therefore,  to  be  brought  about  by  alternate 
southward  component.  In  order  to  lind  the  maximum  me-  currents  in  middle  latitudes  flowing  past  each  other  on  the 
ridional  components  it  is  expedient  to  select  the  following  same  levels,  and  not  over  each  other  at  entirely  different  levels, 
areas  for  the  northward  component:  Low  (16,  8,  2,  7,  15,  6,  as  the  canal  theory  requires.  The  thermal  equilibrium  of  the 
14)  and  High  (18,  10,  11,  19,  12,  20),  and  for  the  southward  air  is,  therefore,  restored  through  the  anticyclonic  and  cyclonic 
component  High  (16,  8,  2,  7,  15,  6,  14)  and  Low  (18,  10,  4,  11,  mechanism,  and  not  by  the  overflowing  currents  from  the 
19,  12,  20).  The  values  of  «,,  r,  are  taken  for  these  areas  from  Tropics  to  the  poles  and  underflowing  currents  from  the  poles 
Tables  9  and  10,  and  the  mean  of  them  is  given  in  Table  12,  to  the  Tropics,  as  commonly  taught.  This  profoundly  modi- 
Northward  and  southward  velocities  in  selected  areas.  It  can  fies  the  canal  theory  of  the  general  circulation  of  the  atmos- 
be  seen  at  once  that  the  general  canal  theory  is  by  no  means  phere  and  introduces  us  to  a  new  point  of  view.  The  discus- 
supported  by  the  observations.  The  fact  seems  to  be  that  sion  of  the  theories  of  the  circulation  of  the  air  as  explained 
between  the  high  and  low  centers,  west  of  the  high  and  east  by  Ferrel,  Oberbeck,  and  other  meteorologists  must  be  taken 
of  the  low,  there  is  a  northward  current  in  all  levels,  strongest  up  next  in  order,  and  their  views  contrasted  with  the  results 
at  about  the  3,000-meter  level,  while  east  of  the  high  and  west  of  our  observations. 


IV.-REVIEW  OF  FERREL'S  AND  OBERBECK'S  THEORIES  OF  THE  LOCAL  AND  THE  GENERAL  CIRCULATIONS.' 


GENERAL    COMPARISON    OF    FERREL  's    AND    OBERBECK'S    THEORIES. 

In  order  to  discuss  the  theories  which  have  been  proposed 
to  account  for  the  circulation  of  the  atmosphere  in  cyclones 
and  anticyclones,  and  in  general  over  an  hemisphere  of  the 
earth,  it  will  be  convenient  to  confine  our  attention  to  the  j 
views  propounded  by  Ferrel  and  Oberbeck  because  their  treat- 
ment is  quite  complete,  and  also  because  they  represent  a  large 
number  of  writers  who  agree  with  them  more  or  less  perfectly. 
There  is  another  theory  of  quite  a  different  type  which  can  be 
taken  up  profitably  after  some  critical  remarks  have  been  made 
on  the  validity  of  these  earlier  views.  In  their  treatment  of 
the  general  circulation  of  the  atmosphere  both  Ferrel  and 
and  Oberbeck  adopt  the  "canal  theory  "  of  the  circulation,  and 
work  out  their  solutions  along  that  line.  Oberbeck  places  his 
origin  of  coordinates  at  the  center  of  the  rotating  earth,  de- 
velops the  equations  of  motion,  and  transforms  to  the  surface 
when  they  are  employed  in  the  evaluation  of  the  resulting  ve- 
locities. He  also  deduces  the  terms  in  the  pressure  due  to 
the  absolute  motion  of  the  earth,  and  to  the  relative  motions 
of  the  atmosphere.  Ferrel  places  his  origin  of  coordinates  at 
the  siirface  of  the  earth,  transforms  his  equations  to  include  a 
temperature  term  through  the  variations  of  the  density,  and 
discusses  the  meaning  of  the  equations  under  special  limita- 
tions, with  illustrations  from  the  observed  motions  of  the 
atmosphere.  It  may  be  remarked  that  von  Helmholtz  intro- 
duces the  temperature  terms  into  the  equations  of  motion,  not 
through  the  density,  but  through  the  pressure,  by  using  the 
equation  of  elasticity,  p  v  =  R  T.  This  procedure  is  probably 
a  better  method  of  solution.  There  is  not  much  difference  in 
the  results  as  derived  from  the  analysis  by  these  authors,  but 
there  is  serious  difficulty  in  making  them  agree  with  the  modern 
observations  of  the  motions  of  the  atmosphere  in  the  higher 
strata,  as  determined  by  the  international  cloud  work. 

In  their  treatment  of  the  cyclone  Ferrel  and  Oberbeck  di- 
verge radically  from  each  other,  though  they  start  with  the 
same  physical  principle,  namely,  a  local  overheated  mass  of  air 
which  in  rising  through  its  own  buoyancy  produces  the  cy- 
clonic circulation.  Ferrel  assumes  a  fixed  cylindrical  boundary 
about  his  cyclone,  and  considers  a  warm-center  cyclone  (circu- 
lation anticlockwise),  surrounded  by  a  pericyclonic  ring  (circu- 
lation clockwise),  in  the  Northern  Hemisphere,  the  two  portions 
being  separated  by  a  surface  where  the  gyratory  velocity 
vanishes.  By  maintaining  a  cold  mass  of  air  in  the  center  in- 
stead of  a  warm  mass  the  circulation  is  reversed,  and  a  cold- 
center  cyclone  is  developed.  Oberbeck  does  not  assume  any 
external  boundary  to  the  circulating  mass  of  air,  but  in  the 
central  region,  bounded  by  a  cylindrical  surface,  there  is  a 
vertical  component,  while  outside  of  it  there  is  no  such  vertical 
ascent  of  the  air.  At  this  boundary  there  is  a  discontinuity  in 
the  vertical  velocity,  and  at  the  same  distance  from  the  center 
the  gyratory  velocity  about  the  axis  is  a  maximum;  this  falls 
away  to  zero  at  the  center  and  also  at  some  indefinite  distance 
in  the  outer  region.  It  is  essential  to  the  existence  of  these  two 
theories,  although  they  differ  so  radically  from  each  other,  to 
establish  the  fact  that  such  local  centers  of  heated  air  in  the 
warm-center  cyclones  do  occur  in  nature,  for  without  them 
these  two  theories  entirely  fail  of  applicability  to  our  meteor- 
ology. They  are  both  possible  forms  of  vortex  motion,  but  il 
is  necessary  to  show  that  the  antecedent  physical  conditions, 
prevail,  before  they  can  be  accepted  as  explanations  of  the  ob- 
served cyclonic  motions. 

IVoni  the  Monthly  Weather  licvicw  lor  April,  l'M'2. 


F.  H.li. 


Both  of  these  authors  have  experienced  much  difficulty  in 
accounting  for  the  anticyclones.  Ferrel  explained  that  the 
interference  of  two  of  his  pericyclonic  rings  would  heap  up  the 
air  and  produce  an  area  of  high  pressure  with  a  clockwise  out- 
flow, but  this  theory  is  so  far  from  being  in  conformity  with 
the  facts,  that  it  is  now,  by  general  consent  of  meteorologists, 
considered  to  be  of  only  historical  value.  Oberbeck  sought,  by 
simply  reversing  the  sign  of  the  vertical  component  of  velocity, 
to  invert  his  cyclone  into  an  anticyclone.  He  met  with  a 
stumbling-block  in  the  mathematical  analysis,  but  was  relieved 
of  this  by  Pockels,  who  correctly  evaluated  the  constant  of 
integration.  No  attempt  was  made  to  show  that  the  resulting 
stream  lines  conform  to  the  motions  of  the  air  in  high  areas 
of  pressure.  Indeed,  since  the  modern  observations  have  given 
us  more  correct  lines  of  flow,  it  is  quite  certain  that  the  anti- 
cyclone can  not  be  explained  in  this  way. 

THE  SUPPLY  OF  LOCAL  CENTERS  OF  HEAT. 

It  is  evident,  therefore,  that  the  first  practical  question  to 
decide  is  whether  such  local  masses  of  air  exist,  heated  in  the 
under  strata  and  more  or  less  cylindrical  in  form,  as  will  pro- 
duce either  of  the  above  forms  of  cyclone.  Meteorologists 
have  generally  supposed  that  this  is  the  case,  and  they  have 
usually  attributed  the  source  of  the  vertical  convection  to  the 
latent  heat  of  condensation.  Dr.  J.  Hann,  in  1890,  and  again 
in  his  Lehrbuch  der  Meteorologie,  has  shown  in  great  detail 
the  inadequacy  of  this  source  of  heat  to  produce  cyclones,  and 
he  has  indicated  that  the  source  of  cyclonic  action  consists 
rather  in  horizontal  convection  currents.  As  this  agrees  with 
the  view  which  I  have  already  advocated,  since  it  seems  to  me 
to  be  in  conformity  with  the  observations,  I  will  therefore 
make  a  resume  of  my  remarks  on  this  topic  in  the  Interna- 
tional Cloud  Report.  It  will  be  a  great  gain  if  meteorologists 
can  be  persuaded  to  reject  the  old  condensation  theory,  which 
has  an  apparent  but  really  illusory  plausibility,  in  favor  of  the 
really  efficient  source  of  dynamic  action  contained  in  the  long, 
horizontal  currents  which  flow  between  the  Tropics  and  the 
polar  regions  in  the  middle  strata  of  the  atmosphere,  as  illus- 
trated in  the  preceding  Paper  III. 

There  is,  in  fact,  a  fundamental  difficulty  in  accounting  for 
the  local  supply  of  heat  which  is  assumed  to  set  the  vertical 
convection  in  operation.  Ferrel  himself  doubted  the  efficiency 
of  the  latent  heat  of  condensation,  for  he  says  in  his  Meteoro- 
logical Researches,  Appendix  No.  10,  United  States  Coast  and 
Geodetic  Survey,  Part  II,  page  201  :  "  The  condensation  of 
aqueous  vapor,  therefore,  plays  an  important  part  in  cyclonic 
disturbances,  but  is  by  no  means  a  primary  or  a  principle 
cause  of  cyclones."  Professor  Loornis  asserted  in  Silliman's 
Journal,  July,  1877:  "Rainfall  is  not  essential  to  the  formation 
of  areas  of  low  barometer,  and  is  not  the  principle  cause  of 
their  formation  or  of  their  progressive  motion."  Indeed,  a 
reasonable  familiarity  with  the  United  States  weather  maps 
proves  conclusively  that  there  are  many  deep,  fully-developed 
storms  which  form  near  the  north  Pacific  coast  and  advance 
to  the  Gulf  of  St.  Lawrence  without  any  precipitation  worth 
mentioning.  Also,  cyclones  form  frequently  in  the  southern 
Rocky  Mountain  districts  and  advance  into  the  lower  Missis- 
sippi Valley  without  any  important  rainfall;  from  that  region 
onward  in  their  course  the  precipitation  and  intensity  of  the 
storm  often  greatly  increase,  since  the  latent  heat  derived  from 
the  inflowing  moist  air  of  the  Gulf  of  Mexico  undoubtedly  assists 
the  vertical  convection  in  the  center  of  the  cyclone.  If  the 

27 


28 


horizontal  currents  which  converge  upon  a  cyclonic  center  are 
bearers  of  moisture,  the  vertical  motion  caused  by  the  dynamic 
action  condenses  the  aqueous  vapor;  but  if  such  currents  are 
drv,  the  cyclone  advances  unattended  by  precipitation.  Hence, 
it  follows  that  rainfall  is  a  secondary  phenomenon,  and  is 
not  sufficient  of  itself  to  produce  true  cyclonic  gyrations. 
There  are,  on  the  other  hand,  many  cases  of  copious  pre- 
cipitation without  any  attendant  low  pressure.  Thus,  on  the 
front  of  an  advancing  cold  wave  there  is  often  a  long  baud  of 
rain  area  stretching  from  the  Great  Lakes  to  the  Gulf  of 
Mexico,  but  without  cyclonic  formation,  the  precipitation  being 
in  fact  caused  by  the  upward  lift  of  a  warm  southerly  current 
which  overflows  the  wedge-shaped  cold  wave  in  its  southward 
movement.  This  is  a  dynamic  uplift  by  overflow,  instead  of  by 
vortical  gyration,  and  it  is  sufficient  to  cause  condensation  and 
precipitation  by  the  mechanical  action  of  an  underflowing 
stratum  of  very  low  temperature.  Furthermore,  on  one  side 
of  a  mountain  range,  as  the  Alps,  rainfall  is  observed  to  oc- 
cur in  the  midst  of  the  high  pressure,  while  on  the  other  side 
of  the  mountains  the  atmosphere  is  clear  and  the  pressure  is 
relatively  low,  thus  reversing  the  required  conditions.  In  the 
summer  season,  local  thunderstorms  are  quite  as  likely  to  hap- 
pen in  the  midst  of  an  area  of  high  pressure  as  in  that  of  low 
pressure,  but  here  the  vertical  convection  distinctly  exists  and 
arises  from  a  superheating  of  the  lower  strata.  If  buoyancy 
of  the  lighter  air  is  the  principal  cause  of  the  gyration  of 
cyclones,  then  we  should  expect  to  find  a  similar  rotatory 
motion  developed  in  the  formation  of  cumulus  clouds  and 
thunderstorms  in  hot  summer  weather,  when  the  vertical  com- 
ponent is  evidently  strong.  But,  on  the  contrary,  while  the 
ascension  of  the  heated  air  is  clearly  visible  in  these  clouds, 
there  is  usually  no  evidence  of  gyration  of  the  cyclonic  type. 
It  has  been  found  by  Hann's  mountain  observations  and  by 
the  Berlin  balloon  ascensions  that  the  temperature  of  the  cen- 
tral portions  of  the  cyclone  is  colder  than  the  temperature  in 
the  midst  of  the  anticyclone  at  the  same  levels.  Hence,  if  the 
relative  density  of  the  air  column  is  the  source  of  cyclonic 
gyration,  we  perceive  that  this  fact  is  in  direct  contradiction  to 
the  requirements  of  the  condensation  theory,  which  demands 
that  the  central  column  of  the  cyclone  shall  be  warmer  than 
its  surroundings. 

Since  the  advocates  of  the  condensation  theory  of  cyclones 
usually  regard  the  generation  of  the  tropical  hurricanes  as  the 
best  example  of  that  source  of  gyratory  energy,  it  may  be 
proper  to  state  that  the  observed  facts  do  not  appear  to  sus- 
tain the  theory.  For  (1)  there  is  no  evidence  of  a  decided 
increase  in  the  local  temperature  at  the  center  of  hurricanes. 
In  this  connection  it  is  believed  that  the  sudden  rise  in  tem- 
perature in  the  Manila  hurricane  of  October  20,  1882,  was  due 
to  the  direct  radiation  of  the  sun  through  the  calm,  eye  of  the 
storm;  (2)  the  winds  are  not  sufficiently  changed  in  direction 
at  the  feeble  ring  of  high  pressure  to  conform  to  the  Ferrel 
pericy clone;  they  should  be  turned  through  at  least  90°  more 
in  azimuth;  (3)  the  conditions  of  heated,  saturated  air  pre- 
vail in  the  Tropics  throughout  the  year,  but  the  hurricanes 
are  produced  at  certain  seasons  only,  and  these  are  the  times 
when  the  counter  currents  of  the  trades  are  most  active  at 
their  northern  and  southern  limits.  Dr.  Hann  rejects  the  rain 
theory,  and  adopts  the  counter  current  theory  for  huricaues: 
Lehrbuch  der  Meteorologie,  pp.  5G3-566.  It  can  be  proved 
conclusively  from  observations  that  two  counter  currents  flow 
together  at  the  places  where  tornadoes  are  formed,  where  the 
tropical  hurricanes  are  generated,  and  also  where  the  cyclones 
of  the  middle  latitudes  are  produced.  These  currents  are 
especially  active  in  the  strata  one  or  two  miles  above  the 
ground,  and  this  is  probably  the  reason  why  they  have  not 
received  due  attention  in  constructing  the  theory  of  storms. 
It  may  be  concluded  that  the  local  overheated  central  region 
does  not  exist  in  cyclones  as  the  chief  cause  of  their  motion, 
and  that  the  theories  fail  which  depend  upon  it.  There  are, 


however,  other  serious  difficulties  of  a  mathematical  nature  to 
which  attention  must  be  directed. 

KEKKEL'S  LOCAL  CYCLONE. 

On  page  595,  and  following,  of  the  International  Cloud 
Report,  the  fundamental  foruiuhe  and  assumptions,  as  em- 
ployed by  Professor  Ferrel  in  his  discussion  of  the  local  c\  clone, 
are  summarized,  and  an  abstract  for  our  purposes,  in  the  nota- 
tion already  described  in  Paper  II  of  this  series,  MONTHLY 
WEATHEU  REVIEW,  February,  1902,  p.  81,  is  given  in  the  follow- 
ing lines: 

Cylindrical  equations  of  motion  applicable  to  the  local  cir- 
culation in  cyclones.  See  International  Cloud  Report,  p.  502. 
1  OP  'In 

185-        —  -      - 


_]  or       <ha 
p  dt    "  <lt  ' 

Assumptions  made  in  duscussing  these  equations: 

1.  The  temperature  is  a  function  of  m  only,  varies  iilong  the 
radius,  but  not  with  the  altitude,  and  is  symmetrical  about  the 
center. 

2.  The  local  cyclone  is  symmetrical  about  the  axis  of  gyra- 
tion, and  is  bounded  by  a  cylindrical  surface  whose  constant 
radius  is  mo. 

3.  The   friction  is  proportional  to   the  relative  velocity  of 
two  adjoining  strata. 

4.  The  assumed  law  of  the  variation  of  temperature  along 
the  radius  is  as  follows,  the  isotherms  being  circles  about  the 
center: 

1  m 

t=    Ao+    gtf,  —  O*08®"*- 

5.  In  integrating  for  the  law  of  the  preservation  of  areas  it 
is  assumed  that  there  is  no   friction  between   the  air  and  the 
surface  of  the  earth. 

G.  All  forces  depending  upon  the  vertical  velocity  of  tin- 
currents  can  be  neglected,  w  =  0. 

7.  Po  =  the  pressure  for  ha  =  0. 

The  equations  of  motions  become  the  following  by  applying 
these  assumptions  and  transforming  the  pressure  term: 


(2.)  0 , 

where  a  =  — 


-  + 


cos  0  -f        ii  + 


,  by  Table  64,  23;  <c  =  tempera- 
ture at  the  center;  and  <u=  temperature  at  the  outer  boundary 
of  the  cyclone. 

As  the  result  of  the  discussion  of  these  two  equations  Ferrel 
deduces  his  cyclone  which  is  represented  in  fig.  11.  The  cor- 
responding cold-center  cyclone  is  shown  in  fig.  12. 

The  first  of  the  above  assumptions  regarding  the  distribu- 
tion of  the  local  temperature  does  not  sufficiently  conform  to 
the  data  on  the  weather  maps  to  be  satisfactory,  because  the 
southeast  section  of  a  cyclone  in  the  United  States  is  usually 
much  warmer  than  the  northwest  section.  The  symmetrical 

1    01' 
distribution  of  pressure  about  the  center,  where  —  -  =0, 

is  found  in  highly  developed  cyclones,  and  may  be  admitted 
in  the  analysis.  The  friction  term  is  of  minor  importance  with 
respect  to  the  general  theory  of  a  cyclone  which  we  are  con- 
sidering, and  the  vertical  force  derived  from  c-  may  be  neg- 
lected, though  not  the  vertical  velocity  itself. 


29 


There   are  two   entirely  different  methods  of   treating   the 
second  equation  of  motion, 

397  b.    (2.)         '^  +  ( 2rt  cos  0  -f  ^\  u  +  IT  =  0, 

and  this  is  the  parting  of  the  ways  between  (1)  Ferrel's  theory 
and  (2)  the  German  theory.  The  primary  question  to  be  kept 
in  mind  is,  does  the  result  of  the  observations  conform  exactly 
to  either  of  these  theories?  This  equation  can  be  integrated 
by  omitting  the  friction  term  ku  and  axxiyniiir/  tin  nuler  boundary; 
or  it  may  be  solved  by  a  simple  transformation,  m'tice  tun  rootn 
run  In-  fon nit,  and  the  discussion  of  the  group  of  general  equa- 
tions of  motion  carried  forward  with  these.  The  former  method 
is  Ferrel's,  and  the  latter  is  that  of  the  German  school,  namely, 
Guldberg  and  Mohu,  Sprung,  Oberbeck,  Pockels,  and  others. 

FERREL'S  SOLUTION. 
Neglecting  the  friction  term,  the  equation  397ft  (2)  can  be 

i)v> 
transformed,  by  substituting   u  =     „.  and  multiplying  by  m, 

into 

rim  dv          dm 


2  n  cos  n  .  m  —  -  +  w  -^  -f  v 


(It 


dv      Ov       udv        vdv 
It  should  be  noted  that  <u  =  ^  +  .)w  +  ^  . 

dv      dv 
Feri-el  integrates  as  if    .    =     .-,  omitting  the  two  other  terms  ; 

indeed,  Ferrel  was  neglectful  about  the  distinction  between 
the  total  and  the  partial  differentials  in  several  portions  of 


(  n  cos  0  -f  —  J  =  c,  for  each 


his  work.     The  integration  gives 

particle.     Assigning  an  outer  boundary  ma  as  the  limit  of  in- 
tegration, then,  for  the  entire  rotating  mass,  we  deduce, 

n  cos  n  -\-    -)••_.  w2  n  cos  0;  and  hence, 


"-(fo-1)  *»«»*. 


where  v  =  the  tangential  velocity  at  the  distance  w  from  the 
axis  of  rotation.  If  i;  =  0,  //  =  0.707  wo,  where  R  is  the  radius 
of  the  circle  at  which  the  gyratory  velocity  reverses  its  direc- 
tion. The  locus  of  this  li  is  indicated  on  fig.  11  for  the  warm- 
center  cyclone  and  on  fig.  12  for  the  cold-center  cyclone;  these 
two  figures  also  show,  in  a  general  way,  the  circulation  in  this 
type  of  vortices.  It  will  not  be  ijecessary  to  explain  it  further 
in  this  connection,  but  it  is  especially  important  to  observe  that 
Ferrel  came  to  this  vortex  by  the  demands  of  his  integration, 
and  that  he  sought  to  uphold  it  by  resorting  to  such  physical 
sources  of  energy  as  seemed  to  be  available.  He  had  already 
applied  an  entirely  similar  process  to  his  discussion  of  the  cir- 
culation of  the  atmosphere  of  the  earth  over  an  entire  hemis- 
phere, but  in  that  case  it  was,  at  least  in  part,  justified  by  the 
fact  that  the  air  on  the  hemisphere  continues  to  remain  the 
same  mass,  so  that  integration  between  the  pole  and  the  plane 
of  the  equator  was  a  proper  procedure.  Yet,  in  comparing 
this  vortex  with  the  circulation  as.  displayed  in  figs.  6  and  7 
of  Paper  III,  we  must  consider  the  other  objections  besides 
the  difficulty  of  accounting  for  local  supply  of  heat  in  the  cen- 
tral portions  which  is  needed  to  keep  the  vortex  in  motion. 

(1)  Ferrel  conceived  the  general  cyclone  on  the  hemisphere  to 
be  one  with  a  cold  center,  since  the  poles  are  cold  and  the 
Tropics  warm;  and  then  the  modification  was  made  that  the 
local  cyclone  is  one  with  a  warm,  center  with  the  edges  cooler 
than  the  middle  portions.  If  a  quantity  of  water  be  placed  in 
a  cylindrical  vessel,  and  sawdust  or  some  other  material  be 
scattered  in  it  to  show  the  lines  of  the  circulation,  and  if  this 
be  rotated  on  a  turntable,  a  form  of  motion  can  be  produced 
quite  similar  to  the  one  Ferrel  proposed  to  explain  the  mechan- 
ism of  storms.  This  circulation  can  be  generated  by  any  agency 


which  will  make  a  vertical  current  in  the  center  of  the  fluid, 
as  a  lamp  on  the  lower  side,  or  a  paddle  screw  at  the  top.  A 
lump  of  ice  on  the  center  of  a  rotating  plane  will  give  a  circu- 
lation which  is  like  that  of  the  general  cyclone  over  the  hem- 
isphere. Now  this  experiment  is  open  to  at  least  three  objec- 
tions of  a  very  serious  nature  when  it  is  attempted  to  apply 
the  lines  of  the  model  to  the  processes  in  nature.  It  is  not 
enough  to  show  that  there  is  an  inflow  at  the  bottom  and  an 
outflow  at  the  top,  in  logarithmic  spiral  curves,  to  conclude 
that  the  analogue  is  satisfactory.  Therein  lies  an  assumption 
which  in  fact  begs  the  entire  question.  The  great  difficulty  is 


Isobars  in  warm-center 
cyclone. 


FIG.  11. — Ferret's  circulation  in  warm-center  cyclones. 


Isobars  in  cold-center 
cyclones. 


FIG.  12. — Ferrel's  circulation  in  cold-center  cyclones. 

that  the  circulation  in  the  general  cyclone  consists  of  the  same 
mass  of  air,  which  repeatedly  passes  through  certain  paths  in 
consequence  of  the  boundary  conditions.  The  limited  mass  of 
fluid  in  the  cylindrical  vessel  is  also  the  same  mass  set  in  cir- 
culation, being  bounded  by  the  top  and  bottom  surfaces  and 
the  curved  sides,  corresponding  with  the  ground  and  top  of 
the  moving  air  and  the  plane  of  the  equator.  It  has  by  no 
means  been  shown  that  the  air  concerned  in  the  local  cyclone 
consists  of  the  same  air  moving  over  and  over  again  in  similar 
paths,  and  it  is  first  necessary  to  do  this  in  order  to  establish 
an  analogue  of  that  kind.  The  evidence  from  the  cloud  circu- 
lations proves  that  the  cyclone  is  a  form  of  circulation  of  the 
stationary  type  of  configuration,  through  which  fresh  portions 
of  the  atmosphere  continue  to  stream.  If  such  is  the  case  the 
analogue  described  above  is  inapplicable,  and  the  deductions 
which  have  been  so  commonly  drawn  from  it  are  quite  incorrect. 
(2)  There  is  no  pericyclone  discoverable  in  the  records  based 
upon  many  .storms.  Ferrel  tried  to  show  that  the  high-area 
pressures  observed  on  the  maps  are  the  resultants  of  two  or 
more  overlapping  pericyclones.  But  the  detailed  construction 
shown  in  Charts  15-35  of  the  International  Cloud  Report  gives 
no  support  to  this  form  of  circulation.  (3)  Evidence  of  true 
cyclonic  outflow  at  the  top  at  some  distance  above  the  ground 
is  probably  entirely  lacking.  The  cyclonic  components  of 
Table  10,  paper  III,  prove  that  the  radial  velocity  is  inward 
from  the  ground  to  the  top  of  the  cyclone.  It  is  not  our  pur- 
pose at  this  point  to  explain  the  principles  of  the  circulation 
that  actually  exists,  but  simply  to  indicate  that  the  Ferrel  cy- 
clone, though  perfectly  possible  under  certain  conditions,  is 
not  the  type  which  storms  follow  in  their  construction.  It  is 
certain  that  the  supposed  analogue  between  the  local  and  the 
general  cyclone  is  not  sustained  by  the  evidence,  and  if  the 
observed  movement  of  the  atmosphere  can  be  accounted  for  on 
other  principles,  in  conformity  with  the  observations,  it  will  only 


30 


add  to  the  force  of  the  position  here  taken  that  the  Ferrel  local 
cvcloneis  merely  one  of  many  ideal!  xed  cases.  For  these  reasons 
we  therefore  are  obliged  to  conclude  that  the  Ferrel  cyclone 
by  no  means  conforms  to  the  natural  circulation,  and  need  not 
be  further  considered.  Indeed,  Ferrel's  teaching  regarding 
the  origin  of  cyclones  and  anticyclones  should  be  eliminated 
from  modern  meteorology. 

THE    GERMAN    SOLUTION. 

If  we  make  the  abbreviation  /  =  2>i  cos  0,  retain  the  friction 

term,  and  make  —  =  -  ° ,  thus  rejecting  the  two  small  terms, 

at       fil 

equation  397/>  (2)  becomes: 

dv       uv       .      .    , 
i)t  +  ~m  +  ' 


422 


o. 


437 


Taking  "v  = 

dv 

u 1 

dra  ^ 


<•><•  ;>m 

dm    (H 

UV        . 
'^  +  ^ 


we  obtain 


+  kv  =  0. 


There  are  two  solutions  of  this  equation,  as  shown  on  pages 
598  and  .599  of  the  Cloud  Report,  namely: 

First  solution  (inner).  Second  solution  (outer). 


-i- 


-         -  u. 
k —  c 


v  =  +  r 

k 


c 

m 


'.  *-e  '   2 

These  can  be  expressed  in  two  general  laws: 
(1.)  Parabolic  law.  (2.)  Hyperbolic  law. 


—  =  —  —  =  constant. 
w  2 

V  ).  C 

=  + .       =  constant. 

w       ^  k  —  c      2 


um  =  —  c  =  constant. 


rro  =  -f-  Tc  =  constant. 

K 


These  solutions  are  readily  verified  by  substitution  in  the 
second  equation  of  motion,  397/>,  and  the  two  forms  give  rise, 
respectively,  to  parabolic  surfaces  on  the  inside  of  a  certain 
circle,  and  to  hyperbolic  surfaces  on  the  outside  of  it.  Their 
discussion  is  given  on  page  509;  an  electrical  analogue  is  ex- 
plained on  page  521,  and  they  are  further  illustrated  on  pages 
619  to  622  of  the  International  Cloud  Report.  A  diagram  of 
the  motion  is  shown  in  fig.  13  of  the  present  paper.  The  re- 
sult is  that  there  is  an  outer  region  in  which  there  is  no  ver- 
tical component,  w  =  0,  and  an  inner  region  in  which  there  is 
a  vertical  component  which  increases  with  the  altitude,  w  =  cz  ; 
see  page  621,  Cloud  Report.  These  two  regions  are  separa- 
ted by  a  circle  where  the  tangential  component  velocity,  v,  is  a 
maximum;  the  velocity  of  rotation  falls  away  to  the  center  by 
the  parabolic  law,  and  also  for  an  unlimited  distance  outward 
by  the  hyperbolic  law.  The  inner  region  has  the  isobars  sepa- 
rated from  each  other  by  distances  conforming  to  the  law, 
d,  =  R*  —  nJ(J,  where  It  is  the  radius  of  the  circle  of  maximum 
velocity,  and  ZB(  the  radii  of  the  successive  isobars;  on  the 
outside  the  distances  between  isobars  are  determined  by 

de  =  2  It1  log  -j-1' ;  fig.  13  shows  these  relative  distances  and  ve- 
H 

loci  ties. 

Recalling  the  circulation  depicted  in  Paper  III,  we  are  in- 
duced to  make  the  following  remarks: 

The  theory  common  to  the  German  school  of  meteorologists 
is  founded  upon  the  assumption  of  a  vertical  central  current, 
like  the  electric  current  in  a  wire,  which  generates  the  cyclonic 
circulation  in  the  inner  and  the  outer  parts.  Now,  there  are  a 
series  of  difficulties  and  objections  to  this  view,  when  it  is  at- 
tempted to  apply  it  to  the  observations  of  the  actual  motions 
of  the  atmosphere,  fully  as  serious  as  those  which  have  been 
urged  against  Professor  Ferrel's  theory.  (1)  There  is  110  suf- 
ficient evidence  that  the  vertical  current  is  of  definite  local 


origin  and  powerful  enough  to  influence  the  enormous  cyclonic 
disturbances  extending  horizontally  to  1,000  miles  in  radius. 
These  storms  are  very  shallow  compared  with  their  width,  say 
3  or  4  to  1,000  at  the  greatest  depth.  An  upward  central  cur- 
rent in  a  small  inner  region  of  200  to  300  miles  radius,  even  if 
locally  produced,  could  hardly  cause  the  disturbances  observed 
on  the  weather  maps.  The  chief  difficult}'  has  been  to  show 
that  there  is  any  sufficient  cause  for  the  existence  of  such  a 
current,  and  the  reasons  already  urged  against  that  view  hold 


FIG.  13. — Oberbeck'B  circulation  in  warm-center  cyok s. 

here  also,  namely,  that  the  disturbing  isotherms  are  not  cir- 
cles, but  their  gradients  lie  athwart  the  cyclone,  generally  from 
S\V  to  NE;  that  cyclones  exist  without  precipitation;  that 
rainfall  does  not  necessarily  produce  cyclonic  action;  and  that 
countercurrents  from  two  specific  directions,  as  MY  and  S,  feed 
into  the  cyclone,  which  is  not  sustained  from  a  supply  equally 
distributed  around  a  center.  (2)  The  adoption  of  the  inner 
and  outer  parts  of  the  cyclone  was  due  to  the  supposed  neces- 
sity of  avoiding  infinitely  great  velocities  at  the  center,  if 

»= —    -  and  y  =  -f—  _ ,  as  would  occur  for  small  values  of  m. 
w  km 

It  will,  of  course,' be  necessary  to  show  how  this  can  be  done 
by  some  other  solution.  Even  if  that  is  accomplished  we  find 
still  other  practical  difficulties  in  the  German  solution  hav- 
ing an  inner  and  an  outer  part.  This  requires  a  maximum  rota- 
tional velocity  v  at  the  boundary  m=li.  But  our  observations 
give  no  support  to  this  position  any  more  than  to  Ferrel's 
theory  that  «=0  at  the  boundary  of  the  inner  and  the  outer 
parts.  A  careful  examination  of  our  wind  velocities  shows  that 
they  increase  steadily  from  the  outer  boundary  toward  the 
center,  when  a  surface  of  discontinuity  surrounding  a  calm 
center  suddenly  terminates  the  radial  and  tangential  velocities. 
The  common  occurrence  of  the  central  calm  in  hurricanes  is 
sufficient  proof  of  this  point.  Furthermore,  an  examination  of 
the  cyclonic  components  ((/2,  r,),  Table  10  and  fig.  10,  shows 
that  the  tangential  velocity  u  increases  from  the  outside  toward 
the  center  without  any  tendency  to  fall  off.  Certainly,  in  fore- 
casting, no  one  expects  to  find  the  maximum  velocities  at  200 
or  300  miles  from  the  center.  The  two-part  theory  itself,  al- 
though gradually  reducing  the  velocity  from  a  maximum  at 
the  boundary  R  to  zero  at  the  center,  does  not  explain  the  ex- 
istence of  the  central  calm.  (3)  While  the  differential  equa- 
tion has  two  solutions  which  give  some  aspects  plausible  for 
this  application,  yet  it  is  improbable  that  in  such  processes  of 


31 


nature  as  the  circulation  of  the  air,  there  should  be  more  than 
one  law  actually  in  operation.  That  the  movement  should  sud- 
denly jump  from  one  system  of  forces  to  another  is  quite  un- 
likely, unless  cause  can  be  shown  for  it.  (4)  In  spite  of  skill- 
ful devices  by  which  discontinuity  in  the  rotation  velocity  was 
overcome,  it  is  evident  that  there  still  remains  a  vertical  dis- 
continuity at  the  boundary,  which  becomes  more  and  more 
pronounced  with  the  increase  in  height  above  the  surface, 
since  w=cz.  While  it  is  hardly  possible  to  actually  observe 
the  vertical  components,  yet  the  probabilities  are  that  vertical 
motion  sets  in  as  soon  as  the  isobars  which  surround  the  cy- 
clone are  closed  up,  and  that  all  over  this  area  there  is  a  rising 
current.  It  may  be  laid  down  as  a  principle  that  where  closed 
isobars  exist  there  is  an  ascending  or  a  descending  current, 
according  to  the  direction  of  the  rotation.  Where  the  isobars 
wander  about  without  closing  up,  it  may  be  assumed  that  there 
is  no  rising  or  descending  air.  In  the  case  of  cyclones  this  is 
confirmed  by  the  general  tendency  of  precipitation  to  occur 
over  the  entire  region  of  the  closed  isobars.  The  preponder- 
ance on  the  eastern  side  over  the  western  is  due  to  the  action 
of  the  general  drift  in  the  upper  strata  upon  the  circulation. 

Hence,  we  conclude  that  while  the  Ferrel  and  the  German 
vortices  are  each  possible  and  may  exist  under  certain  condi- 
tions of  boundary  and  distribution  of  heat,  they  do  not  agree 
with  the  cyclonic  and  anticyclonic  circulation  as  given  by  the 
cloud  observations  of  1896-97.  Although  it  is  not  possible  to 
utilize  the  Ferrel  vortex  in  further  developments  because  the 
outer  boundary  is  lacking,  and  though  the  German  vortex,  on 
the  other  hand,  has  apparently  a  closer  application,  yet  even 
here  it  will  be  found  difficult  to  avail  ourselves  of  it  without 
resorting  to  a  modified  method  of  analysis.  We  shall  show, 
in  part  only,  how  this  may  be  accomplished  in  the  following 
papers,  but  the  fact  remains  that  the  atmospheric  circulation 
is  usually  too  complex  to  be  readily  reduced  to  simple  vortex 
motion  of  any  kind.  Hydrodyuamic  theories  of  stream  lines 
must,  on  the  other  hand,  be  employed  on  a  larger  scale  in  the 
meteorology  of  the  future  than  has  been  done  in  the  past. 

KEKREL'S  THEORY  OF  THE  GENERAL  CIRCULATION  OVER  A  HEMISPHERE. 

We  can  only  briefly  mention  the  principles  which  prevail  in 
Ferrel's  and  in  Oberbeck's  solution  for  the  circulation  of  the 
atmosphere  over  a  hemisphere  of  the  earth.  In  this  case  the 
boundaries  are  fixed,  namely,  the  earth's  surface,  the  plane  of 
the  equator,  the  topmost  stratum  of  the  atmosphere,  and  the 
pole  of  rotation.  The  heat  distribution  is  such  that  the  polar 
regions  are  cold  and  the  Tropics  warm.  The  primary  idea 
adopted  in  the  mathematical  analysis  is  that  of  the  so-called 
canal  circulation,  as,  for  example,  when  fluid  in  a  long  vessel 
with  rectangular  sides  is  artificially  heated  at  one  of  its  ends, 
so  that  the  fluid  rises  at  that  end,  falls  at  the  other,  moves  in 
a  horizontal  direction  from  the  warm  end  toward  the  cold  end 
in  the  upper  layers,  but  from  the  cold  end  to  the  warm  end  in 
the  lower  layers.  In  the  same  way  the  atmosphere  is  assumed 
to  rise  at  the  Tropics,  move  northward  in  the  upper  strata,  fall 
in  the  polar  zones,  and  flow  southward  along  the  surface  of  the 
earth.  The  effect  of  the  contraction  of  the  meridians,  together 
with  the  rotation  of  the  earth,  is  to  introduce  a  complex  torque 
effect,  which  causes  the  air  to  flow  rapidly  eastward  in  the 
temperate  zones,  especially  in  the  upper  strata,  and  westward 
in  the  tropical  zones,  especially  in  the  lower  strata.  The  gen- 
eral result  is  shown  on  fig.  14,  for  Ferrel's  solution;  and  on  fig. 
15  the  relative  component  velocities  are  given  for  Oberbeck's 
solution.  These  two  methods  of  solution  have  some  features 
in  common,  and  also  some  of  the  results  agree,  and  yet  there 
is  wide  divergence  in  other  respects,  as  will  be  indicated.  The 
most  conspicuous  feature  for  us  to  note  is  that  a  neutral  plane 
of  velocity  for  the  components  »  along  the  meridian  is  ob- 
tained, where  there  is  no  northward  or  southward  velocity t  but 
above  it  an  increasing  northward,  and  below  it  an  increasing 
southward  velocity  proportional  to  the  distance  from  this  plane. 


We  shall  have  to  compare  this  view  with  the  results  of  the  ob- 
servations as  given  in  the  data  of  the  year  1896-97.  The  main 
featiires  of  Ferrel's  solution  of  the  general  cyclone  are  con- 
tained in  the  following  extracts  from  pages  588-590,  Interna- 
tional Cloud  Report: 

Polar  equations   of  motion  applicable   to  the  general    cir- 
culation over  a  hemisphere: 

uw  v 

%n  +  v)  v  +  -  -  ;  where  v  = 


"  4  8in" 


Assumptions  are  made  precisely  analogous  to  those  for  the 
local  cyclone,  except  that  the  temperature  is  expressed  by  the 
equation, 

t  =  -  At  cos  x  0,  where, 

A  =  8.50°.     A.=  —  1.75°.     A,=  -  20.95°.     As  =  -  1.00°. 
^4=-2.66°. 

The  equations  of  general  motion  take  the  form, 


200. 
1 

1   OP       du 

~  /T  rOO  ~  dt 
OP            dv 

—  COS0 

I  OP       dw 

397a.    _ 


cos  a  (  In 


dv 
0  =  -,- 


kv. 


The  second  equation  admits  of  integration  between  the  pole 
and  the  plane  of  the  equator  for  the  entire  rotating  mass  of 
air,  with  the  resulting  equation  for  the  velocity  v, 


/2       1  \ 

v  =  I  -7;  •  — — -.  —  sin  (I ) 

V  3    sin  H  J 


If  -r=  0,0=54°  44',  and  the  latitude  <p  =  35°  16'  where 
the  velocity  reverses  direction  at  the  surface. 

The  locus  of  v  =  0,  above  the  surface  forms  an  arch  over  the 
equator,  as  shown  in  fig  14.  This  is  analogous  to  the  curve 
R  of  fig.  12  in  the  cold-center  cyclone. 

TABLE  13. — Theoretical  west-east  velocities. 


Latitude  ..'.. 

v  in  miles  per  hour. 

90° 

+     x    eastward. 

80° 

+3807 

70° 

+  1669 

60° 

+  865 

50° 

+  410 

40° 

+  108 

35°   1C,' 

0 

30° 

-  100    westward. 

20° 

-  239 

10° 

—  320 

Equator  0° 

—  346 

Professor  Ferrel  was  governed  in  his  method  of  integration 
by  the  theorem  of  the  preservation  of  areas,  ra  v=  constant, 
depending  chiefly  upon  the  velocity  u  along  the  parallels  of 
latitude,  in  order  that  the  sum  of  the  momenta  might  be  equal 
to  zero,  Imv=0,  for  the  entire  earth,  which  is  a  necessary 
result.  However,  it  led  to  impossible  velocities,  v,  as  shown  in 
Table  13,  where  excessive  westward  velocities  prevail  in  the 
Tropics,  and  enormous  eastward  velocities  in  the  polar  regions. 
If  we  may  assume  that  the  location  of  the  neutral  plane  is  de- 
termined by  the  fact  that  half  the  air  moves  northward  over 
it,  and  half 'the  air  southward  under  it,  then  the  height  of  this 
plane  should  be  about  6  kilometers=3.7  miles  above  the  ground, 
as  given  in  Table  14. 


32 


TAIIM-:   1-1.      Vi'iiicnl  iliniiinitiini  <if  prmxun-. 


II,  'it-Ill. 

I'lV-SIUV. 

B. 

IVr  rent. 

II.'iKllt. 
//. 

1'IVSMII-C'. 

/;. 

IVr  rent. 

km. 

mm. 

km. 

mm. 

0 

7l!l) 

100.0 

8 

280 

3(1.  '.) 

1 

(171 

88.3 

9 

•217 

32.  fi 

2 

591 

77.8 

10  (Cirrus) 

218 

28.7 

3 

523 

68.9 

11 

193 

25.3 

4 

461 

60.7 

12 

170 

22.4 

5 

407 

53.6 

18 

150 

19.8 

6(A.Cu.) 

359 

47.3 

14 

133 

17.5 

7 

317 

41.8 

15 

117 

15.4 

Fio.  14. — Ferrel's  general  cyclone. 

Ferrel  attempted  to  show  that  the  excessive  east-west  ve- 
locities could  be  reduced  to  proper  proportions  by  introducing 
a  coefficient  of  friction,  and  considering  that  the  sum  of  the 
moments  -»>r&  above  the  neutral  plane  must  be  much  greater 
than  the  -mrjc  below  that  plane.  The  excess  of  energy 
Imvk — -mvjf=E,  must  be  used  up  in  overcoming  the  motion 
of  the  atmosphere,  employing  the  term  friction  to  include  the 
forces  that  retard  circulation  by  internal  turbulent  motion,  or 
by  the  action  of  the  adjacent  discontinuous  surfaces  of  the  larger 
streams.  It  is  evident,  however,  even  supposing  this  theory 


S4°- 


"f 


Jfyua-tor  L 

JKerccti 'oxocl.         ^Ta&t-  West .  VeriicaZ. 

Fio.  15. — Oberboek's  component  motions  in  the  general  cyclone. 

correct,  that  this  source  of  retardation  is  by  no  means  sufficient 
to  overcome  the  great  amount  of  energy  which  must  be  con- 
sumed in  motion  to  equalize  the  heat  energy  derived  from  the 
solar  radiation  in  the  Tropics.  Professor  Ferrel  never  executed 


a  complete  integration  involving  all  the  component  equations 
of  motion,  but  merely  discussed  his  several  equations  tinder 
different  relative  conditions,  and  thus  drew  out  of  them  certain 
conceptions  of  the  general  circulation  of  the  atmosphere  which 
it  was  easy  to  show  harmonized  fairly  well  with  many  of  the 
facts  which  were  known  to  him  at  the  time  of  his  study,  now 
nearly  twenty  years  ago.  It  has  become  increasingly  difficult, 
however,  to  believe  that  this  solution  is  really  as  satisfactory 
as  was  then  supposed. 

OBEKBKCK'S  SOLUTION  or  THE  (IKXEHAI,  CIRCULATION. 

Oberbeck  attacked  the  same  problem  by  a  more  complete 
analysis,  and  reached  conclusions  which  in  general  accord 
with  those  of  Ferrel,  but  differ  from  his  in  important  par- 
ticulars. He  subdivided  the  total  pressure  of  the  atmosphere 
into  partial  pressures,  and  deduced  a  series  of  component 
velocities  which  could  be  computed  by  means  of  coefficients 
distributed  at  equidistant  intervals  throughout  the  atmos- 
phere. An  upper  boundary  was  assumed  for  the  atmosphere, 
but  the  solution  was  conducted  in  such  a  manner  that  this 
limiting  stratum,  whose  height  is  //,  could  be  changed  in  rela- 
tion to  the  radius  of  the  earth,  li.  The  equations  of  motion 
were  constructed  for  an  origin  at  the  center  of  the  earth,  while 
Ferrel's  origin  was  on  the  surface,  but  the  two  systems  of 
equations  can  be  shown  to  be  equivalent,  so  that  the  mathe- 
matical starting  point  is  practically  the  same  in  both.  Ober- 
beck held  all  the  components  together  in  one  system,  and 
hence,  by  not  discussing  them  separately,  could  arrive  at  some 
conclusions  which  are  really  more  instructive  than  Ferrel's. 
Yet  it  will  be  easily  seen  that  these  do  not  conform  sufficiently 
well  to  the  data  of  observation  to  be  accepted  as  the  complete 
solution  of  the  problem. 

Taking  the  component  equations  and  notation  given  on 
pages  591-593  of  the  International  Cloud  Report,  Oberbeck  's 
solution  for  the  component  velocities  is  as  follows: 

South: 


it  =  G  6  cos  ft  sin  0   -^  (6/i2  —  15/i*  +  8-r2). 


'48 


2)i 
«,,=  —  7r  cos  <>  sin 

3  K 

East: 


,,2  _  7^  Cos2  II)  j-K  (G  li'—  15/1  ff+ 8-7* 


<,  =  D  sin  0  (1  —  3  cos2  0)  ^  (—  9/r5  +  15/iV  —  15//*4  +  4-r5). 


,  =  D  6  cos2  II  ^  (20/>V  -  25/)^  +  8,r4). 
Zenith: 
=  (<  (i  _  3  cos2  H)  ~  (l>  —  «)(3h«  —  W). 

r* 

0,  +  2va  —  6  (4v,  +  >t)  cos2  H  +  35>,  cos4  #] 
_  ff)(3/i  _  2^.) 


x 

}  -  H 
= 


f  °    A   A       ™ 
=  \TO'  10'  TO  •  •  •  io 


C=  0.5429  It1;  D  =  0.00008532  R\ 

r  =  11  -(-  Rh  =  R  +  H=  radius  of  earth  +  height  of  atmos- 
phere. 

r=n+tt«  =  R  +  1'-  Rh  =  radius  of  earth  +  height  of  the 

stratum  <r;  h  is  the  maximum  value  of  v. 

Assumed  law  of  relative  angular  velocity,  •;  =  •••,  cos2  H  —  *2. 

Tables  15,  10,  17,  give  the  relative  coefficients  of  the  com- 
ponent velocities  in  the  direction  of  the  meridians,  the  paral- 
lels of  latitude,  and  in  the  vertical,  respectively.  By  assign- 
ing values  to  //  the  different  velocities  may  be  computed. 

I  have  not  been  successful  in  obtaining  such  velocities  as 
will  harmonize  at  all  well  with  the  known  movements  of  the 
atmosphere,  and  I  have,  therefore,  been  led  to  distrust  the 


33 


TABLE  15. 

I.   Component*  on  the  meridians  due  to  the  rotation  of  tlte  earth. 
u  =  horizontal  currents  on  meridians.     +  —  south,  —  —  north. 


9 

0° 

10° 

20°                  30°                  45°                  50°                  60°                  70° 

80° 

90° 

._  H3             ^n  H3        (  2.203  X  10'  for  H=63700'". 
Coefficients  of  C  R3  —  .  0429  -R  =  -j  g  20g  £  K)4  for  H_  637() 

(7=1.0 

.0000 

—  .  0213 

—.0401          —.0540          —.0614          —.0614          —.0540          —.0401 

—  .  0213 

.0000 

.9 

.  0000 

—     196 

—     368                  496          —     564          —     564          —    496          —    368 

—     196 

.0000 

.8 

.  0000 

151 

•     283              -     382          —    434          —     434          —     382          —     283 

—     151 

.0000 

.7 

.  0000 

87 

164                  221              .     251           —     251          —     221          —     164 

—      87 

.0000 

.6 

.  0000 

—       15 

—      29          —      39          —       44          —      44          —      39          —      29 

15 

.0000 

.5 

.0000 

+       53 

+     100           +     135           +     154           +     154           +     135           +     100 

+       53 

.0000 

.4 

.  0000 

+     110 

+     206           +     278           f     316          +     316          +     278          +     206 

+     110 

.0000 

.3 

tOOOO 

+     143 

+     268           +     361           +     411          +     411          +     361          +     268 

+     143 

.0000 

.2 

.  0000 

+     142 

+     266          +     359          +     408          +     408           +     359          +     266 

+     142 

.0000 

.1 

.0000 

+       97 

+     183           +     247           +     281           +     281          +     247          +     183 

+       97 

.0000 

.0 

.0000 

.0000 

.0000               .0000               .0000               .0000               .0000               .0000 

.0000 

.0000 

II.   Components  on  the  meridians  due  to  the  relative  motion  of  the  atmosphere. 

«i,  =  horizontal  currents  on  meridians.     +=  south,  —  =  north. 

.  2n         H3                                         \  .  4062  X  105  for  H=63700m. 
Coefficients  of  -  Iff  w  =  .  1571  X  lO'9  H3  =  j  .  4062  X  W  for  H=  6370. 

0=1.0 

.0000 

—  .  0426 

—.0713           —.0782          —.0641          —.0376          —.0112           +.0050 

+.0073 

.0000 

.9 

.0000 

-     392 

_     655          —     718          —     588          —     346          —     103           +       46 

+       67 

.0000 

.8 

.  0000 

—     301 

_    504          —     553          —     452          —     266          —      79          +35 

+       51 

.0000 

.7 

.  0000 

174 

—     292          —     320          —     262          —     154          —      46           +20 

+       30 

.0000 

.6 

.  0000 

—       31 

—      51          —      56          —      46          —      27          —        8           +4 

+         5 

.0000 

.5 

.  0000 

+     107 

+     178           +     196           +     160           +94           +28           —       12 

—       18 

.0000 

.4 

.  0000 

+     219 

+     367           +     402           +     330          +     194          +58          —      26 

—      37 

.  0000 

.3 

.  0000 

+     285 

+     477           +     523           +     428          +     252          +75          —      33 

—      49 

.  0000 

.2 

.  0000 

+     283 

+     473           +     519          +     425          +     250          +75          —      33 

—      48 

.0000 

.1 

.  0000 

+     195 

+     327           +     357           +     293          +     172          +51          —      23 

—       33 

.0000 

.0 

.  0000 

.0000 

.0000                .0000                .0000                .0000                .0000                .0000 

.0000 

.0000 

TABLE  16. 

I.  First  components  on  the  parallels  due  to  the  rotation  of  the  earth. 

D,  =  horizontal  currents  on  parallels.      +  =  east,  —  —  west. 

o=1.0 

.00000 

+.00346 

Re                            R*       j  1.405  X  105  for  £frz6370. 
+  .(KI.-,s'.i         +.00651         +.00509        +.00191         —.00226        —.00636 

—  .  00934 

—  .  01042 

.9 

.00000 

+       345 

+       588        +       650        +       510        +       190        —      226        —      634 

—      932 

—     1040 

.8 

.(»oooo 

+       341 

+       580        +       641         +       502         +       188        —      223        —      626 

—       911) 

.—     1026 

.7 

.00000 

+       328 

+       55;i         _|_       G18         +       484         +       181         —       215         —       603 

—      884 

—      989 

,6 

.  00000 

+       307 

+       522         +       578         +       452         +       169         —       201         —       564 

—      828 

—       924 

.5 

.00000 

+       275 

+       467         +       517         +       404         +       151         —       179                   504 

741 

—       827 

.4 

.  00000 

+       231 

+       395         +       437         +       342         +       128         —       152                   426 

626 

—       699 

.3 

.00000 

+       181 

+       307         +       340         +       266         +       100         —       118                   332 

—       487 

—       544 

.2 

.(10000 

+        123 

+       210        +       232         +181         +     .    68        —        81        —      226 

—      332 

—       371 

.1 

.0(1(100 

+         62 

+       106         +       117         +         91         +         34         —         41         —       114 

168 

187 

.0 

.  00000 

.00000 

.00000             .00000             .00000             .00000             .00000             .00000 

.  00000 

.  00000 

34 


TAJIL.E  16 — ContiiiuiMl. 

II.  Second  components  on  the  parallels  iliif  In  lh<-  rolntioii  of  I  In  i-urlli. 
t>a  — hori/.oiital  currents  on  parallels.       :   —  east,  — —  \vc.-t. 


H 

0° 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

90° 

o=:1.0 

.00000 

+.00315 

+  .00565 

of  Z>^  =  .8 

+  .00702 

+  .  00701V 

r5     ,  i.io.- 

t  —\  1.405 

+  .OO.V.I2 

X  10s   for  //  = 
+  .00101 

-|  .(Xl2or, 

:,ooo:,i; 

.  in  ii  ii  in 

.9 

.00000 

+       305 

+       547 

+       680 

+       683 

+       573 

+       391 

+ 

199 

+        54 

.00000 

.8 

.00000 

+       276 

+       495 

+       614 

+       618 

+       518 

+       354 

+ 

180 

+         111 

1111:11  III 

.7 

.00000 

+       231 

.+       415 

+       515 

+       518 

+       435 

+       297 

+ 

157 

+         41 

.  IHXXMI 

.6 

.00000 

+        179 

+       321 

+       398 

+       401 

+       336 

+       229 

+ 

117 

+          32 

.  01  KM  III 

.5 

.00000 

+       125 

+       225 

+       279 

+       281 

+       235 

+       161 

+ 

82 

+         22 

.OOOOO 

.4 

.00000 

+         76 

+       136 

+       169 

+       170 

+       142 

+         97 

+ 

50 

+          13 

.  IKNXX) 

.3 

.00000 

+         37 

+         67 

+         83 

+         84 

+         70 

+         48 

+ 

21 

+           7 

.IXNXX) 

.2 

.00000 

+         13 

+         24 

+         29 

+         29 

+         25 

+         17 

+ 

9 

+           2 

.00000 

.1 

.00000 

+          2 

+           4 

+           4 

+          5 

+           4 

+           3 

+ 

1 

0 

.  1  II  II  II  II  1 

.0 

.00000 

.00000 

.00000 

.00000 

.00000 

.00000 

.00000 

.OOtXMI 

.  ooooo 

.ooooo 

TABLE  17. 

I.    Vertical  components  due  to  the 

rotation  of  the  earth. 

w  =  vertical  currents. 

+  =  ascending,  —  =  descending. 

IT  4                                 f-fi 

Coefficients  of  C  ^4  —  .  5429  j^- 

|  2.  203  X 
=  }  2.  203  x 

l&toi  #=63700"'. 
10   for  #=6370. 

<r=1.0 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

0000 

.0000 

.0000 

.9 

—.0244 

—  .  0233 

—.0202 

—  .  0152 

—  .  0093 

—.0029 

+  .0030 

+ 

0079 

+  .0111 

+  .0122 

.8 

—    448 

428 

—     370 

—     280 

—     170 

54 

+       56 

+ 

145 

+      201 

+     224 

.7 

—    588 

—    562 

—     486 

—     368 

—     224 

T-      70 

+       74 

+ 

191 

+     268 

+      2'.  U 

.6 

—     648 

.     619 

535 

—     405 

—     247 

—      78 

.  +       81 

+ 

210 

+    2'.ir, 

+      324 

.5 

—     624 

—    596 

—     515 

—     390 

—     237 

—      75 

+       78 

+ 

202 

+     284 

+      312 

.4 

—     528 

—     504 

—     436 

—     330 

•     201 

—      63 

+       66 

+ 

171 

+     210 

+      264 

.3 

—     378 

361 

—     312 

—     236 

144 

45 

+       47 

+ 

123 

+     172 

+      1N9 

.2 

—    208 

—     199 

172 

—     130 

79 

91 

+       26 

+ 

68 

+       95 

+      104 

.1 

64 

—       61 

—      53 

40 

24 

8 

+         « 

+ 

21 

+       29 

+        32 

.0 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

0000 

.0000 

.0008 

II.    Vertical  components  due  to  the  relative  motion  of 

the  atmosphere. 

w3  =  vertical  currents. 

+  =  ascending,  —  =  descending. 

2n        H* 

_«lf*       (.4062V  10"    for  #=63700'". 

Coefficients  c 

f  --  R*  -ft-t  = 

.  1571  X  10 

R   —  1  .  4062  X  10~'  for  #=6370 

<r  =  1.0 

.0000 

.0000 

.0000 

.0000 

.4000 

.0000 

.(XXX) 

(XIIXI 

.  OOIKI 

.  oooo 

.9 

+  .0502 

+  .0444 

+  .0294 

+  .0106 

—  .0050 

—.0128 

—.0119 

—  . 

0053 

+  .0020 

+  .1X15(1 

.8 

+     927 

+     820 

+     542 

+     197 

—      92 

237 

—     220 

— 

97 

+       37 

+        93 

.7 

+  1217 

+  1077 

+     712 

+     258 

—     121 

311 

—    289 

— 

127 

+       49 

+     122 

.6 

+  1341 

+  1187 

+     785 

+     285 

—     133 

—     343 

—     319 

— 

140 

+        53 

+     134 

.5 

+  1294 

+  1145 

+    757 

+     275 

—     129 

—    331 

—     307 

— 

135 

+        52 

+      130 

.4 

+  1093 

+     967 

+     639 

+     232 

—     109 

—     279 

—    260 

— 

114 

+        44 

+      11(1 

.3 

+    782 

+     692 

+     458 

.+     166 

—       78 

—     21X1 

—     186 

— 

82 

+       31 

+       78 

.2 

+     430 

+     380 

+     251 

+       91 

43 

110 

—     102 

— 

45 

+       17 

+       43 

.1 

+     132 

+     117 

+       77 

+       28 

—       13 

—       34 

—       31 

— 

14 

+         5 

+       13 

.0 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

0000 

.0000 

.0000 

validity  of  this  "  canal  theory  ' '  of  the  circulation,  as  explained 
in  niy  report.  An  inspection  of  the  coefficients,  Table  15,  shows 
that  in  the  case  of  the  meridian  components,  u,  the  upper  north- 
ward and  the  lower  southward  circulation  as  deduced  for  the 
rotating  earth,  is  somewhat  modified  by  the  component  u.t  de- 


pending on  the  motion  of  the  atmosphere  relative  to  the  earth 
itself.  The  component  c  along  the  parallels  of  latitude,  Table 
1C),  differs  radically  from  Ferrel's  result,  and  it  should  be  care- 
fully noted.  Oberbeck  divides  the  eastward  drift  in  higher 
latitudes  from  the  westward  drift  in  lower  latitudes  at  the 


35 


parallel  35°.  The  eastward  v  is  a  maximum  in  the  neighbor- 
hood of  latitude  60°,  l>  =  30°,  but  vanishes  at  the  poles.  This 
is  exactly  contrary  to  Ferrel's  result,  which  made  the  velocity 
i'  a  maximum  at  the  pole,  before  the  assumed  modification  by 
friction  was  applied.  Oberbeck  makes  the  westward  drift  a 
maximum  at  the  plane  of  the  equator,  which  is  certainly  not 
in  conformity  with  the  observations.  He  also  makes  the  west- 
ward velocity  increase  at  the  equator  from  the  surface  to  the 
upper  boundary,  and  show  no  sign  of  a  reversal  from  westward 
to  eastward  at  a  moderate  elevation,  as  is  generally  believed 
to  be  the  fact,  judging  from  certain  well  known  motions  of  the 
air  observed  in  the  Tropics. 

The  United  States  Weather  Bureau  has  been  conducting  a 
series  of  uephoscope  observations  in  the  West  Indies  for  the 
past  three  years,  and  it  is  hoped  that  the  discussion  of  these 
observations,  soon  to  be  undertaken,  will  give  us  some  defi- 
nite information  on  this  important  point. 

The  second  term  i\  modifies  ri;  but  the  two  combined,  ti  =  vt 
-f-  i~v  sustain  the  conclusions  just  mentioned.  This  feature  of 
Oberbeck 's  solution  is  so  far  from  conforming  to  the  observed 
motions  of  the  atmosphere  that  it  seems  to  me  to  be  inferior 
in  value  to  Ferrel's  for  the  Tropics.  Ferrel's  arch  over  the 
Tropics,  shown  in  fig.  14,  is  probably  a  fact,  and  if  this  is  so, 
then  the  only  serious  modification  required  in  Ferrel's  treat- 
ment is  to  show  how  the  excessive  eastward  drift  in  the  mid- 
latitiide  and  polar  zones  can  be  effectively  checked.  It  is  evi- 
dent that  there  must  be  a  large  amount  of  energy  available 
for  use  in  the  construction  of  local  cyclones  and  anticyclones, 
and  that  there  is,  therefore,  no  pressing  need  to  refer  the 


energy  of  these  motions  to  any  local  supply  of  heat,  as  is  done 
by  those  who  extend  to  cyclones  the  theory  of  the  latent  heat 
of  condensation  from  precipitation  originally  devised  by  Espy 
to  explain  cumulus  clouds  and  thunderstorms.  The  compo- 
nents w  and  wt,  Table  17,  show  that  there  is  an  ascending 
current  in  the  Tropics,  and  a  descending  current  in  the  higher 
latitudes.  Thus,  as  the  result  of  the  theoretical  discussion  in 
general,  the  canal  theory  has  several  of  its  features  verified, 
and  yet  there  are  serious  discrepancies  inherent  in  both  Fer- 
rel's  and  Oberbeck's  solutions. 

My  statement  has  suggested  by  implication  that  there  exists 
an  important  principle  which  has  been  neglected  by  these  me- 
teorologists. They  have  each  discussed  the  general  and  the 
local  cyclones  as  if  they  were  in  a  sense  independent  of  one  an- 
other, since  separate  sources  of  heat  energy  are  assigned  to 
each,  and  two  characteristic  laws  of  circulation  are  deduced 
therefrom.  It  is  much  more  natural  to  suppose  that  these  two 
systems  are  mutually  interdependent,  and  that  the  excess  of 
energy  of  the  general  cyclone  is  transformed  into  the  driving 
forces  of  the  local  circulation;  also,  that  the  acquired  motion 
of  the  local  cyclone  reacts  upon  and  retards  the  excess  of  mo- 
tion of  the  general  cyclone  in  the  temperate  zones.  The  sub- 
ject becomes,  however,  excessively  complex,  and  I  can  only 
attempt  to  sketch  in  general  terms  in  my  next  paper  an  out- 
line of  this  view,  hoping  some  other  time  to  be  able  to  supple- 
ment it  with  a  more  suitable  mathematical  analysis,  when  the 
study  of  the  observations  now  in  hand  has  been  advanced  more 
nearly  to  completion. 


V.— RELATIONS  BETWEEN  THE  GENERAL  CIRCULATION  AND  THE  CYCLONES  AND  ANTICYCLONES.1 


UNEQUAL  DISTRIBUTION  OF    CYCLONES  IN    NOKTH  AMERICA  AND 

EUROPE-ASIA. 

We  have  arrived  at  the  following  proposition  as  the  result ' 
of  the  discussion  of  Ferrel's  and  Oberbeck's  analysis  of  the 
general  and  local  circulation,  that  the  general  cyclone  and  the 
local  cyclones  and  anticyclones  have  been  treated  almost  inde- 
pendently of  one  another,  while  in  fact  the  imperfect  results 
of  the  theory  and  the  modern  observations  both  indicate  that 
these  two  classes  of  movement  should  be  analyzed  in  close  rela- 
tion with  each  other.  The  evidence  compels  us  to  regard  both 
these  circulations  as  the  common  effect  of  the  readjustment  of 
the  thermal  equilibrium,  which  is  disturbed  by  the  radiation 
of  the  sun  falling  on  the  tropic  zones,  and  the  true  meteoro- 
logical problem  is  to  trace  out  the  successive  stages  in  the 
process  of  this  interaction  through  the  resulting  currents 
which  circulate  in  the  atmosphere.  The  results  contained  in 
this  paper  apply  especially  to  the  North  American  Continent, 
and  it  is  hardly  to  be  expected  that  the  details  will  be  found 
the  same  in  all  the  other  regions  of  the  earth.  Indeed  there 
are  several  reasons  for  believing  that  this  continent  is  the  pe- 
culiar theater  for  the  interchange  between  the  heat  and  the 
cold  of  the  Northern  Hemisphere,  and  that  the  Euro-Asian 
Continent  plays  a  very  different  role  in  the  meteorological 
economy  of  this  hemisphere.  For  it  is  well  known  (1)  that 
while  the  American  Continent  is  the  place  for  the  profuse  gen- 
eration of  cyclones,  Europe  is  the  region  for  their  dissipation, 
and  in  Asia  very  few  cyclones  occur  except  along  the  ocean  bor- 
ders; (2)  that  the  velocity  of  motion  of  the  atmosphere  gener- 
ally is  about  twice  as  great  over  North  America  as  over  Europe. 
This  points  to  a  profound  difference  between  the  actions  of 
the  atmosphere  in  these  two  regions,  but  one  cause  of  it  at 
least  is  easily  perceived.  It  has  been  shown  that  the  currents 
which  are  especially  concerned  in  forming  cyclones  are  con- 
tained for  the  most  part  within  2  or  3  miles  of  the  ground, 
though  their  accompanying  effects  may  extend  much  higher. 
Hence,  any  barriers  of  elevated  ground,  as  mountain  ranges, 
which  tend  to  deflect  the  flow  of  the  lower  strata,  must 
strongly  influence  the  formation  of  the  cyclones  themselves, 
if  they  are  to  be  referred  to  the  counter  flow  of  long  horizontal 
currents  of  different  temperature  rather  than  to  local  vertical 
couvective  currents. 

The  great  range  of  the  Himalaya  Mountains  stretching  east 
and  west  is  such  a  barrier  to  the  flow  of  the  tropical  and  polar 
currents  in  that  region,  and  the  result  is  that  true  cyclonic 
movements  are  almost  excluded  from  the  interior  of  Asia.  On 
the  other  hand  the  Rocky  Mountain  range,  stretching  north 
and  south  along  the  western  districts  of  North  America,  favors 
the  counter  flow  from  the  Tropics  and  the  polar  regions  by  de- 
flecting the  westward  current  of  the  Tropics  toward  the  north, 
and  the  eastward  drift  of  the  higher  latitudes  toward  the  south. 
The  same  tendency  is  favored  by  the  location  of  the  high  pres- 
sure belt  in  the  latitude  of  35°,  which  causes  a  high  pressure 
area  to  form  over  the  middle  Atlantic  Ocean,  while  the  Eocky 
Mountain  range  breaks  through  the  midst  of  it.  The  result  is 
to  produce  a  powerful  anticyclouic  center  of  action  over  the 
Atlantic  Ocean,  which  maintains  a  strong  northward  compo- 
nent from  the  West  Indies  toward  the  interior  of  the  conti- 
nent. At  the  same  time  the  American  Continent  causes  the 
isobars  and  isotherms  to  loop  southward,  and  thus  in  conse- 
quence to  draw  the  Siberian  atmosphere  in  a  direction  nearly 
parallel  to  the  Rocky  Mountain  range.  These  physical  condi- 
tions are  a  constant  incentive  to  the  formation  of  coimtercur- 


rents  which  meet  on  the  Canadian  and  United  States  territory, 
with  the  result  that  75  per  cent  or  80  per  cent  of  the  storms  of 
the  Northern  Hemisphere  are  generated  in  these  districts.  It 
is  not  necessary  for  maintaining  the  temperature  equilibrium 
of  the  hemisphere  that  the  interchange  of  heat  and  cold  should 
occur  so  as  to  have  a  uniform  distribution  over  all  portions  of 
it,  because  if  there  is  an  excessive  interchange  in  any  place,  as 
in  North  America,  the  general  movements  of  the  atmosphere 
will  soon  transfer  the  effects  to  all  other  parts.  Keeping  these 
facts  in  mind  will  facilitate  an  understanding  of  the  views 
which  will  be  briefly  described  as  follows: 

CRITICISM    OF    THE    CANAL    THEORY    OF    THE    GENERAL    CIRCULATION. 

The  immediate  problem  before  us  is  this:  To  what  extent  is 
the  canal  theory  of  the  general  circulation  over  a  hemisphere 
correct,  and  in  what  direction  must  it  be  modified  to  conform  to 
the  modern  observations?  Perrel  derived  the  following  equa- 
tions from  a  discussion  of  the  first  equation  of  397a  for  the 
approximate  velocities  and  gradients  in  the  strata  of  the  upper 
atmosphere : 

0.016       ah 
408a.    Velocity; 


409a.    Gradient;  G=  G 


(2n  + v)  '  r 
0.00001327 


At  sin  0 

(1  +  *)  '' 
A,  sin  20 


when  u0  G. 


P0are 


the 


(1  +  at)'    P9     ' 
values  at  the  surface,  and  v,  G,  P,  the 


1  Koj>rintc<l  I'mm  the  Monthly  Weather  llovicw  for  May,  11)02. 


values  at  the  height  h.  .  Since  y!2is  negative,  —  20.95°,  it  fol- 
lows that  v  is  greater  than  rc,  and  G  is  greater  than  G0,  so  that 
the  eastward  velocity  and  the  meridional  gradient  velocity  in- 
crease with  the  height.  The  relative  eastward  velocity  is 
shown  on  fig.  1(5.  The  interchanging  velocity  along  the  meri- 
dians is  northward  above  the  neutral  plane  and  southward  be- 
low it  as  given  on  fig.  16. 


N 


-U. 


Fici.  16. Fi'rrol's  component  currents  by  the  canal  theory. 

The  canal  theory  requires  that  the  flow  of  the  atmosphere 
should  be  in  unbroken  stream  lines  conforming  to  these  pre- 
cepts. The  result  of  the  principle  of  preservation  of  areas, 
•o  a  =  constant,  as  applied  to  the  axis  of  rotation  of  the  earth, 
is  that  the  velocity  v  is  excessive  near  the  poles,  and  the  gra- 
dients of  the  upper  strata  much  too  large.  Ferrel  sought  to 
escape  this  difficulty  by  evaluating  -m  (v  —  va)  k  for  the  upper 
strata  with  a  northward  —  it,  and  -m  (v—  DO)  k  for  the  lower 

37 


38 


strata  with  a  southward  +  u,  and  assuming  that  the  difference 
of  momenta  is  equal  to  the  retardation  of  the  eastward  drift 
by  the  frictioual  resistance.  It  is  known,  however,  that  the 
frictional  coefficient  is  a  very  small  factor,  and  not  capable  of 
producing  the  required  retardation. 

Furthermore,  the  modern  observations  show,  as  in  Paper 
III,2  that  the  northward  —  «,  and  the  southward  +  M,  compo- 
nents are  not  distributed  as  the  canal  theory  requires,  but 
that  there  are  approximately  equal  currents  flowing  northward 
and  southward  on  the  *«///<•  levels,  which  reduces  the  difference 
of  momenta  to  zero,  and  is  fatal  to  the  frictional  theory  of 
retardation.  The  counter  flow  of  horizontal  currents  on  the 
same  levels,  most  powerful  in  the  strato-cumulus  level,  does, 
however,  constitute  a  dynamic  mechanism  quite  capable  of 
retarding  the  eastward  drift,  so  as  to  produce  the  observed 
moderate  eastward  velocities  of  the  temperate  zones.  This  is 
the  true  source  of  the  energy  consumed  in  the  motions  of 
cyclones  and  anticyclones.  Compare  fig.  17. 


FIG.    17. — Bigelow's  component    currents  from  the  Weather    Bureau 

observations. 

The  general  theory  which  has  already  been  mentioned  in 
my  previous  paper — Storms,  Storm  Tracks,  and  Weather  Fore- 
casting, Bulletin  No.  20,  Weather  Bureau,  1897 — is  practically 
as  follows:  There  is  a  sheet  of  the  atmosphere  flowing  quite 
steadily  eastward  over  the  United  States  at  the  levels  from  2 
miles  upward.  Beneath  this  flows  north  and  south,  quite 
independently  of  the  upper  drift,  a  series  of  counter  currents 
which  are  cold  and  warm,  respectively.  The  interaction  of 
these  currents  sets  up  the  anticyclonic  and  cyclonic  whirls 
chiefly  by  dynamic  action,  the  former  causing  a  downward 
component  and  the  latter  an  upw.ard  component.  The  cyclone 
receives  its  supply  from  the  northwest  and  from  the  south, 
and  thus  discharges  great  masses  of  air  through  itself,  so  that 
the  cyclonic  configuration  constitutes  a  type  of  persistent  cir- 
culation in  a  vortical  form  of  stream  lines.  This  gyratory 
rotation  lifts  the  air  from  one  level  to  another,  purely  by  its 
mechanical  action,  and  thus  raises  the  mass  from  the  strata 
of  lower  to  those  of  higher  eastward  velocity.  This  rapid 
change  of  eastward  motion,  if  imparted  to  the  air  raised  up, 
can  take  place  only  by  an  interchange  of  inertia,  that  which 
is  gained  by  the  rising  air  being  lost  by  the  upper  eastward 
drift.  Also  the  gyration  of  the  lower  strata  causes  deflection 
in  the  eastward  drift  by  the  composition  of  forces,  as  must  be 
the  case,  since  the  method  followed  in  our  discussion  has  been 
to  subtract  the  normal  west-east  and  north-south  velocities 
from  those  actually  observed,  the  residuals  being  the  circular 
cyclonic  movements  that  are  now  described. 

The  point  of  view  for  the  consideration  of  the  theory  of  the 
general  cyclone  is  thus  considerably  altered.  (1)  Instead  of 

•'Monthly  WV;il  II.T  Kr\i.>\v,  Miiivli   liin-J,  Vol.  XXX,  p.  117. 


a  circulation  as  in  a  canal,  northward  above  and  southward 
below,  we  find  that  the  interchanging  motion  is  largely  con- 
fined to  the  lower  strata,  by  means  of  currents  not  flowing 
above  one  another,  but  on  the  same  level.  (2)  Instead  of  the 
momentum  M  (i'o  —  <•)  being  determined  by  the  difference  be- 
tween the  eastward  flow  at  different  levels  of  the  same  latitude, 
we  prefer  the  statement  tluit  cyclonic  gyrations  are  produced 
by  the  counter  flow  of  independent  streams,  and  that  the  rapid 
eastward  drift  is  retarded  by  mechanical  inflows,  from  above 
toward  the  base  of  the  anticyclone,  and  from  below  through 
the  stream  lines  of  the  cyclone  into  the  eastward  drift.  The 
energy  upon  which  cyclones  and  anticyclones  depend  for  their 
activity  is  to  be  traced  to  a  different  source  from  that  generally 
assigned  to  it  by  meteorologists.  The  common  theory  is  that 
the  cyclone  is  due  to  some  form  of  vertical  convection,  caused 
by  overheating  a  local  region,  and  by  the  latent  heat  produced 
in  precipitation.  Our  theory  would  more  naturally  depend 
upon  horizontal  convection,  by  means  of  which  temperature 
gradients  thousands  of  miles  in  extent  produce  comparatively 
shallow  streams,  which  flow  from  the  north  and  from  the  south, 
and  sustain  them  for  considerable  intervals  of  time.  The  up- 
ward and  the  downward  discharges,  together  with  the  rotary 
components  which  make  the  sinuous  flow  of  the  air  in  the 
upper  levels,  practically  tie  together  the  upper  and  the  lower 
strata,  retard  the  eastward  drift  in  the  higher  strata,  and 
accelerate  the  eastward  motion  in  the  lower.  This  effect  is 
readily  perceived  in  the  eastward  propagation  of  high  and 
low  pressure  areas  over  the  United  States,  which  is  the  basis 
of  our  system  of  forecasting  and  renders  it  possible. 

It  is  by  no  means  to  be  concluded  that  by  suggesting  this 
modification  of  the  Ferrel  theory,  any  intention  exists  of  not 
recognizing  fully  the  fact  that  it  remains  substantially  correct 
in  some  of  its  features.  There  exists  the  eastward  drift 
throughout  the  middle  latitudes  and  the  westward  drift  in 
lower  levels  of  the  tropical  zone.  But  there  is  yet  another 
reservation  to  be  introduced.  The  heating  of  the  tropical  belt 
raises  the  isobars  adjacent  to  the  equator  so  that  they  slope 
toward  the  poles,  and  to  such  an  extent  that  they  almost  ex- 
actly counterbalance  the  deflective  force  2/iy  cos  tl,  which  is 
directed  southward.  It  may  properly  be  assumed  from  this 
that  in  the  upper  strata  the  directions  of  the  isobars,  the  iso- 
therms, and  the  stream  lines  of  the  wind  motion  are  verv 
nearly  parallel  to  each  other,  if  not  coincident.  The  friction 
is  evidently  small,  judging  from  theoretical  conditions,  and 
from  the  results  of  the  observations,  or  else  this  could  not  be 
the  case. 

There  is  another  important  deduction  to  be  drawn  from  this 
discussion  regarding  the  flow  of  currents  from  the  Tropics 
toward  the  poles  in  the  lower  strata.  According  to  the  Ferrel 
theory,  the  overheating  of  the  Tropics  is  relieved  by  the  up- 
ward expansion  and  overflow,  but  in  accordance  with  the  pres- 
ent view  the  tropical  congestion  is  relieved  by  irregular  streams 
which  flow  outward  from  the  lower  levels.  This  being  the 
case,  the  poleward  gradients  in  the  upper  levels  are  called 
upon  to  sustain  much  less  pressure  from  the  deflection,  and 
evidently  the  tendency  to  excessive  eastward  velocities  is  much 
diminished,  just  because  the  equatorial  lift  of  the  strata  can 
not  be  so  great,  since  the  expansion  upward,  as  stated,  leaks  off 
sidewise  in  the  lower  levels.  The  eastward  drift  does  not 
therefore  increase  to  excessive  values  for  these  two  reasons: 
(1)  The  tropical  strata  are  not  elevated  up  to'  the  theoretical 
amount  because  of  the  escape  of  the  currents  poleward  not 
very  far  from  the  surface  of  the  ground;  (2)  the  eastward  drift 
is  diminished  by  the  operation  of  the  vertical  discharges  be- 
tween the  lower  and  the  higher  levels  produced  by  the  purely 
mechanical  vortex  motion  in  the  cyclones  and  the  anticyclones. 

The  evidence  before  us  is  to  the  effect  that  the  heating  of 
the  atmosphere  is  generally  confined  to  a  layer  less  than  5,000 
meters,  or  3  miles,  thick.  It  is  not  intended  to  allude  now  to 
the  annual  range  in  temperature  with  the  sun's  change  in 


39 


latitude,  but  rather  to  tlie  shorter  periods  of  only  a  few  days 
length  which  contribute  to  the  impulse  of  streams  from  the 
south.  There  are  several  reasons  which  lead  to  this  conclu- 
sion: (1)  The  trend  of  the  preceding  argument  has  been  to 
show  that  the  readjustments  of  disturbed  temperature  equi- 
librium take  place  in  the  lower  layers  of  the  atmosphere  by 
means  of  rather  spasmodic  impulses,  controlled  partly  by  the 
temperature  energy  in  the  tropic  and  the  polar  regions,  partly 
by  the  distribution  of  land  and  ocean  temperatures,  and  bv 
the  relative  radiation  which  takes  place  from  them  in  the 
winter  and  summer  season,  respectively.  (2)  Half  of  the 
mass  of  the  atmosphere  is  contained  below  the  5,000-meter 
level,  and  this  is  the  layer  within  which  the  greater  part  of 
the  aqueous  vapor  is  also  collected,  since  the  vapor  contents 
of  the  higher  levels  is  in  the  form, of  fine  ice  crystals,  which 
drift  eastward,  encircling  the  earth,  and  perhaps  seldom  find- 
ing their  way  to  the  ground.  The  fact  that  the  dust  of  the 
Krakatoa  volcano  was  thus  carried  about  the  earth  for  two  or 
three  years  shows  that  the  upper  current  has  a  history  of  its 
own,  to  a  considerable  extent  independent  of  the  3-inile  layer 
nearest  the  earth.  Now,  the  important  part  which  the  aqueous 
vapor  plays  in  the  absorption  of  heat  is  well  understood,  and 
it  depends  upon  the  very  high  latent  heat  of  water,  which  is 
606.5  calories  per  kilogram  at  0°  C.  The  evaporation  and 
condensation  of  water  in  precipitation  is  certainly  confined  to 
the  lower  stratum,  and  hence  it  is  in  harmony  with  this  view 
to  limit  the  effective  heating  of  the  air  by  the  sun  to  the  lowest 
3  miles.  (3)  The  diurnal  variation  of  the  temperature  at  the 
surface  of  the  ground  takes  on  a  wide  range.  The  tempera- 
ture is  above  the  normal  in  low  areas,  in  the  summer  season, 
and  in  the  middle  of  the  day;  it  is  below  the  normal  in  high 
areas,  especially  in  the  case  of  cold  waves  in  the  winter,  and 
in  the  night-time.  The  range  at  the  ground  generally  amounts 
to  10°  or  20°  F.,  but  diminishes  upward  with  the  height  and 
disappears  at  the  5,000-foot  level,  or  even  considerably  below 
that  height.  This  is  shown  very  clearly  in  the  study  of  the 
changes  in  the  vertical  temperatures,  as  explored  by  means  of 
balloon  ascensions,  where  the  5,000-foot  level  marks  the  con- 
vergence of  the  lines  which  represent  the  gradients  in  the 
forenoon  and  afternoon.  The  range  in  the  United  States  is 
iisually  greater  than  in  Europe  at  the  ground,  owing  to  the 
more  pronounced  nature  of  the  cold  and  warm  waves  that 
move  eastward  over  that  region,  but  the  evidence  is  that  the 
diurnal  lines  converge  at  2  or  3  miles  above  the  ground.  (4) 
The  fact  that  the  great  eastward  drift  of  the  upper  levels  is 
underruu  by  a  series  of  comparatively  thin  currents,  which 
move  about  in  every  possible  direction,  shows  that  the  dis- 
turbance of  equilibrium  is  local  and  confined  to  a  shallow  skin 
near  the  ground,  the  most  rapid  currents  belonging  to  the 
cumulus  and  strato-cumulus  levels,  which  also  implies  that  the 
upper  regions  of  the  atmosphere  are  much  less  affected  than 
the  lower.  (5)  The  same  conclusion  is  indicated  very  clearly 
by  the  seasonal  change  in  the  drift  of  the  high  areas  over 
the  northwestern  portions  of  the  United  States,  from  the 
northwest  in  winter  and  from  the  southwest  in  summer,  in 
conformity  with  the  location  of  the  permanent  high  areas  in 
winter  over  the  continent  and  in  summer  over  the  ocean. 

From  these  considerations  it  seems  evident  that  the  upper 
atmosphere  is  but  slightly  disturbed  in  its  temperature  equi- 
librium by  the  effects  of  the  solar  radiation,  but  the  solar  rays 
puss  through  it'with  comparatively  little  absorption,  while  the 
larger  percentage  of  the  heat  retained  in  the  lower  strata,  is 
due  to  a  change  of  the  wave  length.  This  conclusion  is 
very  important  for  two  reasons:  (1)  It  shows  why  the  gen- 
eral circulation  of  the  atmosphere  prescribed  by  the  Ferrel- 
Oberbeck  theory  does  not  seem  to  be  confirmed  by  the  ob- 
servations. (2)  It  also  indicates  where  the  energy  comes 
from  which  is  finally  expended  in  the  generation  of  cyclonic 
circulations  and  in  the  retardation  of  the  eastward  drift  by 
the  agency  of  inertia  rather  than  by  friction;  for  if  the  total 


amount  of  energy  falling  upon  the  Tropics  does  not  expend 
itself  in  an  upper  poleward  current,  because  the  higher  strata 
retain  a  temperature  of  equilibrium  almost  undisturbed,  then 
this  energy  must  give  rise  to  a  series  of  comparatively  small 
currents  moving  poleward  in  the  lower  strata — a  fact  which 
is  abundantly  confirmed  by  the  observations.  Also,  if  the 
friction  of  the  upper  atmosphere  upon  itself  is  very  slight, 
then  there  will  be  but  little  retardation  to  an  excessive  east- 
ward drift,  tending  under  a  steady  force  to  have  a  constant 
acceleration.  The  only  other  available  agency  which  will  pro- 
duce the  same  effect  is  the  intrusion  into  the  higher  strata 
from  the  lower,  or  vice  versa,  of  air  moving  eastward  with  a 
different  velocity,  which  must  suddenly  be  subject  to  accelera- 
tion. The  discharge  of  the  product  passing  through  cyclonic 
circulations  is  perfectly  fitted  to  perform  this  office.  Hence, 
the  theory  here  expounded  consists  of  two  parts  as  regards 
the  eastward  drift:  (1)  The  upper  poleward  gradients  are  not 
built  up  to  the  amounts  supposed  by  Ferrel,  because  of  the 
lateral  escape  poleward  of  currents  from  the  tropical  belt; 
(2)  The  agency  of  friction  as  a  retarder  is  replaced  by  the 
interchange  of  inertia  derived  from  a  compound  circulation, 
the  sources  of  the  separate  parts  having  different  and  inde- 
pendent causes.  It  has  been  important  to  thus  carefully  clear 
the  ground  for  the  theory  of  local  cyclones,  which  will  be  ad- 
vanced to  take  the  place  of  the  type  proposed  by  Professor 
Ferrel  on  the  one  hand,  or  of  that  advocated  by  the  German 
school  on  the  other  hand.  These  two  theories  are  not  in 
harmony  with  each  other,  and  neither  of  them  seems  to  be  in 
agreement  with  the  observations.  It  is  very  evident  that  the 
six  assumptions  which  have  been  made  in  order  to  pass  from 
the  general  equations  of  motion,  200,  to  the  working  system 
employed  by  Ferrel,  397a,  must  be  carefully  revised  before  we 
can  expect  to  put  this  branch  of  meteorology  upon  a  correct 
working  basis.  In  particular  it  is  not  suitable  to  omit  the 
variations  of  temperature  in  longitude,  because  in  so  doing 
the  turbulence  of  the  lower  strata  and  the  alternate  streams 
which  are  implied  in  this  variation  profoundly  modify  the 
general  and  the  local  circulation.  It  is  also  necessary  to  cor- 
rect the  statement  regarding  the  friction  as  the  special  agency 
of  retarding  accelerated  flows  and  substitute,  or  rather  add 
thereto,  the  inertia  of  currents  which  are  rapidly  changing 
their  velocities.  The  conflict  of  turbulent  countercurrents, 
especially  at  a  short  distance  above  the  ground,  must  be  rig- 
orously considered  in  studying  the  resultant  effects.  Like- 
wise the  direct  application  of  the  law  of  conservation  of  areas 
passed  over  by  the  rotating  radius  vector  can  not  apply  imme- 
diately to  the  lower  strata,  though  it  may  be  much  more  nearly 
correct  above  the  3-mile  level. 

MODIFICATION    OF    THE    CANAL    THEORY. 

In  consequence  of  these  considerations  it  seems  necessary 
to  modify  the  canal  theory  to  such  an  extent  as  to  be  practi- 
cally equivalent  to  an  abandonment  of  it.  If  this  is  done  it  is 
important  to  trace  out  a  chain  of  circumstances  which  will  give 
a  more  correct  account  of  the  general  and  the  local  circulation 
of  the  atmosphere.  The  canal  theory  is  very  artificial,  depend- 
ing as  it  does  upon  a  simple  laboratory  experiment,  in  connec- 
tion with  an  obvious  analysis  of  the  general  equations  of  mo- 
tion. If  this  theory  is  to  be  preserved,  then  we  must  be  assured 
that  the  atmosphere  does  in  fact  traverse  the  circuits  pre- 
scribed, for  it  has  been  commonly  assumed  to  conform  with 
the  facts  of  observation.  As  has  been  shown,  observation  does 
not  bear  out  the  requirements  of  the  theory,  and  I  shall,  there- 
fore, attempt  to  trace  out  in  a  descriptive  way  the  circulation 
as  it  is  developed  over  the  North  American  Continent.  The  re- 
duction of  this  kinematic  picture  to  the  corresponding  mathe- 
matical form  of  dynamics  is  a  task  of  very  great  difficulty,  as 
may  readily  be  inferred. 

We  may  conceive  the  tropical  strata  to  be  elevated  by  ther- 
mal expansion  relative  to  the  polar  regions,  so  that  there  is  a 
certain  average  gradient  slope  and  corresponding  west-east 


40 


velocity  which  is  in  equilibrium  with  it  in  accordance  with  the 
usual  ('(illations.  If  the  canal  theory  of  circulation  is  in  opera- 
tion we  have  poleward  gradients  in  the  upper  strata,  and 
equatorward  gradients  in  the  lower  strata.  If,  instead  of 
maintaining  this  circulation,  there  is  an  escape  of  currents 
from  the  Tropics  in  the  lower  strata  in  a  poleward  direction, 
then  the  gradients  of  the  canal  theory  diminish  and  become 
much  more  moderate  in  consequence  of  this  release  of  ten- 
sion. Every  such  leakage  current  from  the  Tropics  causes  a 
break  or  fault  in  the  gradient,  and  this  must  be  attended  by 
a  corresponding  deflection  of  the  eastward  current,  through  the 
operation  of  the  deflecting  force  at  right  angles  to  it,  due  to 
the  earth's  rotation.  Referring  to  the  scheme  of  self-regulation 
of  the  circulation  by  the  rise  and  fall  of  the  gradients,  tig.  18, 


FIG.  18.— Scheme  of  self-regulation  of  the  circulation  by  the  rise  and  fall 
of  the  gradient. 

we  may  consider  the  average  gradient  in  the  strato-cumulus 
level.  When  in  its  mean  position,  as  determined  by  the  whole 
set  of  natural  circumstances  which  produce  it,  the  isobars  and 
the  wind  with  a  given  direction  and  velocity  are  all  practically 
in  coincidence.  The  velocity  is  just  enough  to  maintain  the 
slope  of  pressure  as  measured  by  the  gradient.  If  the  thermal 
expansion  of  the  Tropics  increases,  the  gradient  slope  is  ele- 
vated, the  eastward  velocity  is  increased,  and  this  acceleration 
will  continue  to  advance  till  an  excess  of  energy  secures  for 
itself  a  way  of  escape.  This  increase  of  velocity  is  probably 
checked  as  follows: 

From  the  Tropics  in  the  lower  levels,  that  is  in  the  cumulus 
and  strato-cumulus  strata,  a  stream  of  air  breaks  away  from 
the  canal  circuit,  and  pushes  northward  in  some  irregular 
course.  This  evidently  causes  a  break  in  the  gradient  sur- 
faces, such  as  is  indicated,  for  example,  by  the  dotted  lines  of 
fig.  18,  and  the  slope  of  pressure  which  held  the  eastward 
drift  in  its  position  gives  way.  The  action  of  the  deflecting 
force  due  to  the  earth's  rotation  bends  the  current  southward 
as  is  indicated,  and  there  is  thus  made  the  beginning  of  an  an- 
ticyclonic  local  movement.  In  the  lowest  levels,  from  cumulus 
down  to  the  surface,  the  break  in  the  average  gradients  may  be 
so  pronounced  as  to  offer  but  little  check  to  a  complete  anticy- 
clonic  gyration,  such  as  appears  at  the  surface  of  the  earth. 
This  deflection  of  direction  causes  a  whirl  and  absorption  of 
energy  in  the  strata  affected  through  the  local  interchange  of 
inertia,  and  slows  up  the  eastward  drift  by  the  vortex  action 
which  is  produced  with  a  downward  component.  The  alter- 
nate rise  and  fall  of  the  gradients,  and  the  attendant  anticy- 
clonic  gyrations,  mark  the  successive  efforts  at  self-regulation 
which  the  atmosphere  as  a  thermal  engine  imposes  upon  itself. 
These  continental  pulsations  are  shown  on  the  weather  maps 
as  the  procession  of  high  and  low  pressure  areas  which  traverse 
the  middle  latitudes.  In  consequence  of  the  superior  velocity 
of  eastward  motion  in  the  strato-cumulus  level,  this  region  is 
the  first  to  feel  the  decline  in  gradient  strength,  due  to  the 
tendency  of  a  stream  to  escape  from  the  Tropics,  as  from  the 
Gulf  of  Mexico  over  the  United  States.  It  thus  happens  that 


the  anticyclone  is  not  only  larger  in  area,  lint  it  also  generally 
precedes  the  cyclone  in  its  formation.  There  are  numerous 
instances  in  which  the  anticyclono  overspreads  the  United 
States,  while  there  is  no  important  cyclone  in  connection  with 
it,  except  possibly  some  depression  or  irregular  action  along 
the  edges  of  the  high  area.  Usually,  however,  incipient  cy- 
clones increase  in  intensity  from  these  small  beginnings,  and 
they  may  even  seem  to  drink  up  and  exhaust  the  air  which  is 
flowing  in  anticv  clonic  circulation.  This  is  the  reason  why  I 
have  heretofore  described  the  anticyclone  as  preceding  the 
cyclone  in  efficiency,  and  have  thus  reversed  the  order  of 
action  as  taught  by  Professor  Ferrel  in  his  well-known  theorv. 
The  warm  stream  from  the  south  is  deterred  from  mingling 
with  the  cold  antievclonic  air  by  the  difference  of  its  tempera- 
ture, and  the  result  is  that  the  eastern  side  of  the  anticyclone, 
which  flows  southward,  and  the  warm  escape  current  from  the 
south,  flowing  northward,  compose  two  counter  currents. 
These  two  currents  together  generate  the  cyclone,  which  is  a 
vortex  with  an  ascending  central  velocity.  The  gyration  is 
produced  by  the  action  of  the  two  independent  streams  acting 
like  a  couple,  since  they  each  depend  upon  separate  gradient 
systems  for  their  own  mechanical  pressures.  It  is  not  a  g\  ra- 
tion due  to  frictional  impact,  but  rather  to  the  steady  pressure 
on  the  arms  of  a  couple,  since  the  streams  are  driven  by  inde- 
pendent gradients,  and  are  held  apart  by  having  different 
temperatures  in  the  two  separate  currents.  The  air  is  thus 
raised  from  the  lower  to  the  higher  strata  in  great  masses,  by 
circulating  through  the  configuration  of  the  cyclone,  and  this, 
too,  produces  a  retardation  of  the  eastward  drift  by  an  imme- 
diate interchange  in  the  inertia,  since  there  is  a  quick  mingling 
of  air  having  different  directions  and  velocities.  Examples  of 
this  action  can  be  seen  by  studying  the  Charts  '20  to  35,  in- 
clusive, of  the  International  Cloud  Report.  Furthermore,  the 
motions  of  the  atmosphere  result  in  placing  strata  of  air  ha\  ing 
different  temperatures  together,  side  by  side,  so  that  the  sur- 
face isotherms  are  directed  from  the  southwest  to  the  north- 
east, and  in  consequence  the  northwest  portions  of  a  cyclone 
are  cold  and  the  southeast  portions  are  warm.  This  does  not 
conform  to  the  requirements  of  either  the  Ferrel  or  the  Ger- 
man theories,  which  demand  a  warm  local  center  for  the  gen- 
eration of  cyclones.  Yet  if  two  such  masses  of  air  lay  along- 
side while  they  are  of  different  temperatures,  an  interchange! 
of  heat  contents  will  take  place  locally  between  them,  and  thus 
the  streams  will  interflow  in  such  a  way  as  to  strengthen  the 
cyclonic  gyration.  This  will  be  accompanied  by  distinct  strati- 
fication of  the  air  currents  in  the  local  cyclones,  such  as  is 
observed  in  the  kite  and  balloon  ascensions,  since  the  air  of 
different  temperatures  is  drawn  out  into  thin  ribbons  having 
large  discontinuous  surfaces,  which  are  favorable  to  the  inter- 
change of  heat.  It  is  frequently  found  that  a  great  anticy- 
clonic  area  as  it  approaches  the  ocean,  althoiigh  attended  by 
no  cyclone,  will  yet  suddenly  cause  a  violent  whirl  to  form  on 
its  edge  by  the  mere  action  of  these  adjacent  masses  of  dif- 
ferent temperatures.  Such  local  storms  are  sometimes  formed 
on  the  Atlantic  coast  during  a  single  night,  and  they  may  cause 
vortices  with  hurricane  velocities  on  the  coast.  The  line  of 
junction  of  these  warm  and  cold  currents,  along  the  southern 
and  southeastern  parts  of  the  low  area,  is  the  locus  of  the 
formation  of  the  majority  of  the  tornadoes  of  the  United 
States,  the  counter  flow  setting  up  the  gyration,  which  is  con- 
verted into  a  genuine  columnar  vortex,  through  which  the 
heated  air  of  the  lower  strata  escapes  into  the  colder  strata  of 
the  higher  levels.  The  hurricanes  of  the  "West  Indies  simi- 
larly form  along  the  places  of  the  counter  flow  between  the 
Atlantic  high  area  and  the  southeast  trade  winds  when  at 
their  extreme  northern  limit,  as  in  August  to  October.  The 
vortex  then  travels  westward  and  skirts  around  the  periphery 
of  the  high  area  until  it  is  absorbed  finally  in  the  eastward 
drift  of  the  higher  latitudes.  Such  dynamic  intermingling  of 
the  general  and  the  local  circulation  is,  therefore,  not  only 


41 


in  accordance  with  observations,  but  it  is  a  suitable  substi- 
tute for  tlie  defective  canal  theory  of  the  general  circulation, 
and  also  for  the  untenable  theory  of  the  local  cyclones  and 
anticyclones,  supposed  to  be  dependent  upon  the  central  heat 
produced  by  condensation  of  the  aqueous  vapor  of  precipita- 
tion. This  view  is  attended,  on  the  other  hand,  by  the  follow- 
ing disadvantage:  That  while  the  canal  theory  and  the  warm 
center  cyclone  theory  lend  themselves  readily  to  mathematical 
treatment  and  to  analytic  solutions  of  considerable  elegance, 
we  are  obliged  to  substitute  for  them  an  irregular  system  of 
stream  lines  in  the  lower  strata,  not  at  all  readily  put  into 
mathematical  forms.  This  turbulent  circulation,  with  its  self- 
adjusting  government  of  the  eastward  flow,  its  interaction  be- 
tween the  general  and  the  local  vortices,  its  numerous  subordi- 
nate phenomena,  such  as  tornadoes,  hurricanes,  and  cyclones,  is 
easy  to  comprehend,  but  hard  to  analyze  mathematically  into 
the"  exact  dynamic  forces  of  equilibrium.  It  is  possible  to 
construct  several  special  typical  configurations  for  each  dis- 
trict of  the  earth,  as  was  done  for  the  United  States  in  Charts 
20-35,  inclusive,  of  the  International  Cloud  Report,  and  then 
draw  the  stream  lines,  with  their  velocities,  in  order  to  prepare 
for  the  computation  of  the  dynamic  forces  involved.  This  is 
the  true  meteorological  problem  of  the  future. 

THE    STRUCTURE    OF    THE    ANTICYCLONE. 

We  will  now  examine  a  little  more  closely  the  structure  of 
the  anticyclone  and  the  cyclone  as  given  by  observations  for 
the  sake  of  the  analytical  problems  presented  by  their  configu- 
ration. It  has  been  claimed  by  meteorologists  that  there  is  a 
southward  component  in  the  middle  strata  of  the  north  temper- 
ate zone  toward  the  high  pressure  belt,  but  a  northward  compo- 
nent in  the  upper  strata  and  another  northward  component 
near  the  surface,  as  is  indicated  on  fig.  13,  Paper  IV.  The  obser- 
vations of  1896-97  do  not,  however,  give  such  a  distribution  oi 
the  mean  components,  for  they  show  that  there  is  a  very  small 
average  drift  northward  in  all  strata,  increasing  slightly  with 
the  height  above  the  surface.  That  is  to  say,  the  atmosphere  in 
the  eastern  and  central  United  States  drifts  northward  a  very 
little  and  thus  supplies  part  of  the  air  that  descends  into  the 
anticyclones  through  the  upper  strata.  We  have  indicated  how 
the  leakage  in  the  lower  strata  from  the  Tropics  in  part  re- 
places the  air  which  descends  in  the  anticyclonic  areas,  and  it 
is  assumed  that  the  small  residual  northward  drift  comple- 
ments the  amount  that  is  required  to  fill  the  anticyclonic 
areas.  The  downward  vortex,  therefore,  draws  in  a  portion  of 
the  air  passing  through  it  from  the  upper  strata,  as  a  conse- 
quence of  the  gyration  induced  through  the  countercurrent 
action,  and  therefore  the  feeble  northward  component  of  the 
circulation  of  the  higher  strata  seeks  the  surface  practically 
in  the  middle  latitudes,  before  arriving  at  the  polar  zone 
through  the  mechanism  of  the  local  vortices.  Hence,  it  fol 
lows  that  there  is  little  cause  for  the  formation  of  a  genera 
anticyclone  close  to  the  pole  itself,  which  Ferrel  assumed  to 
exist;  the  result  of  Kimball's  discussion  in  the  MONTHLY 
WEATHER  REVIEW,  September,  1901,  goes  to  show  that  thii 
movement  only  feebly  exists,  and  is  in  conformity  with  thi; 
exposition  of  the  general  circulation.  Therefore,  the  air  tha 
descends  in  a  local  anticyclone  comes  from  two  sources,  the 
leakage  currents  from  the  Tropics  in  the  lower  and  middl 
strata  and  the  feeble  northward  drift  in  all  strata,  especially 
the  higher. 

There  is  one  feature  of  the  anticyclonic  vortex  which   may 
be  mentioned,  though  it  belongs  more  properly  to  an  analyti 
treatment  of  that  circulation.     The  anticyclonic  component 
of  fig.  6,  Paper  IIP,  show  that  we  are  not  dealing  with  a  pur 
form  of  vortex.     The  two  possible  laws  are  typically  the  para 

bolic  —  =  constant,  and  the  hyperbolic  vat  =  constant. 


» Monthly  Weather  Koview,  April,  1902,  Vol.  XXX,  p.  166. 
'Monthly  Weather  Review,  March,  1902, Vol.  XXX,  p.  117. 


Hyperbolic. 
m= constant. 


Parabolic. 

=  constant. 


FIG.  19.— Mixed  system  of  hyperbolic  and  parabolic  components. 
According  to  the  parabolic  law  —  =  constant,   the    circula- 

ion  causes  simply  a  depression  in  the  center  of  the  gyrating 
.uid;  according  to  the  hyperbolic  law  i:w  =  constant,  there  is  a 
ertical  component  of  circulation  as  in  ordinary  vortices.  In 
he  observed  anticyclonic  components  the  velocities  v  are  about 
qual  to  each  other  on  the  I,  II,  III,  circles,  and  it  seems  to 
ne  that  this  can  only  happen  if  there  is  a  mixture  of  these 
wo  laws  of  motion.  Thus  we  may  divide  the  observed  com- 
ionents  rI:  ru,  i'm,  into  two  parts  by  a  diagonal  as  shown  on 
ig.  19.  The  upper  components  represent  the  parabolic,  and 
he  lower  the  hyperbolic  portions.  This  is  physically  neces- 
ary  for  the  following  reason.  The  anticyclone  is  formed  by 
arge  currents  of  air  moving  in  more  or  less  independent 
treams  on  its  outer  portions,  while  only  curling  offshoots 
•each  its  central  parts;  this  would  produce  the  pure  para- 
)olic  components  only.  But,  through  imperfect  pressure 
gradients  there  is  also  near  the  center  a  true  downward  com- 
ponent of  circulation,  and  this  can  be  supplied  only  by  the 
iction  of  hyperbolic  components,  that  is  to  say,  of  a  simple 
vortex  motion.  Hence,  the  general  anticlockwise  movement 
f  the  anticyclone,  strongest  on  the  outer  circles,  has  accom- 
panying it  a  true  downward  or  vortex  component  which 
engthens  the  components  u  in  the  central  portions.  If  this  is 
correct,  one  sees  an  additional  reason  for  holding  that  Fer- 
el's  explanation  of  the  anticyclone  is  impracticable,  and  also 
that  the  reversing  of  the  cyclone  to  make  an  anticyclone,  as 
proposed  by  Oberbeck  and  Pockels  is  not  warranted.  We  need 
rery  accurate  observations  to  settle  so  difficult  a  point  of  pure 
;heory,  but  I  can  not  at  present  see  any  other  satisfactory  ex- 
planation of  the  gyratory  components  derived  from  the  Weather 
Bureau  observations. 

STRUCTURE  OF  THE  CYCLONE. 

In  Table  18  we  give  the  results  of  cloud  observations  in  the 
United  States. 

In  the  following  example  the  relations  which  should  exist  in 
a  pure  vortex  are  deduced  for  comparison  with  the  data  given 
under  low  areas  in  Table  18. 

Taking  the  inward  radial  velocity  u  =  — 1.25,  at  the  distance 
1,250  kilometers,  assuming  A  =  .000100  for  #  =  4(5°  17',  and 
c  =  .000002,  also  k  =  .000050,  their  introduction  into  the 
several  formulse  gives  the  values  found  in  Table  19. 

An  account  will  be  given  of  the  derivation  of  the  formulae 
in  Paper  VI. 

It  is  seen  that  the  rotational  velocity  -u  is  about  the  same  as 
that  given  by  the  observations  up  to  the  circle  whose  radius  is 
150  km  =  93  miles  from  the  center,  as  seen  under  \\  of  Table 
18.  The  values  of  the  radial  velocity  agree  fairly  well,  if  we 
admit  that  the  observations  may  be  somewhat  imperfect  for 
this  component  up  to  the  region  of  the  inner  circle  I,  whose 
radius  is  250  km.  There  the  component  u  is  much  larger  than 
expected  in  the  upper  strata,  and  this  indicates  some  opposi- 
tion to  the  free  development  of  the  vortex  near  the  core.  It 
is  presumed  that  this  implies  a  struggle  to  intrude  into  the 
rapid  eastward  drift,  accompanied  by  a  broadening  of  the  vor- 
tex tube  through  the  resistance,  v  being  smaller  than  it  should 
be  and  u  greater,  making  the  angle  of  the  inclination  i  much 
greater  than  it  ought  to  be  in  pure  vortex  motion.  As  the 
computation  of  the  vortex  goes  to  the  height  of  10  miles,  far 
beyond  the  altitude  to  which  the  ordinary  cyclonic  motion 
penetrates,  this  being  only  3  or  4  miles  in  the  moderate  move- 
ments of  the  air,  it  is  seen  that  v  should  theoretically  attain 


42 


enormous  velocities  very  near  the  center.  The  difference  be- 
tween those  and  such  as  are  actually  observed  may  be  regarded 
as  measuring  the  energy  expended  in  breaking  up  the  cyclone 
in  the  higher  levels,  which  can  be  balanced  only  by  retarding 
the  movements  of  the  general  cyclone.  The  vortical  velocity 
w  is  extraordinarily  small  from  the  bottom  to  the  top,  and  in 
a  measure  justifies  the  method  of  discussing  the  motions  of 
the  air  in  the  cyclone  as  a  case  of  horizontal  movement.  It  is 
impossible  theoretically,  however,  that  such  a  cyclonic  motion 


without  a  vertical  component  should  exist  at  all.  The  fact  is 
tli at  a  slow  vertical  movement  is  acting  over  the  very  large 
area  covered  by  the  cyclone,  and  is  sufficient  to  carry  oft'  all 
(lie  air  which  flows  into  it  through  a  thin  disk  at  the  outside 
in  horizontal  directions  toward  the  center.  The  checks  m  v 
and  in  w  =  —  2  u  s  hold  throughout  the  cyclone,  thus  proving 
that  our  data  are  at  least  approximately  correct.  The  angle 
(',  between  the  tangential  direction  and  the  current,  theoret- 
ically becomes  very  small  within  300  miles  of  the  center.  It 


TABLE  18. — Anticyclonic  and  cyclonic  velocities  at  each  l,(WO-metf,r  level.     (Copy  of  Table  126  International  Claud  Report.) 


Cloud  forms. 

Height  in 
meters. 

High  areas. 

Low  llrcav. 

Velocities  in  Hi,' 
k'rnrntl  cycloiu-. 

«1        »l 

I 

«2                  «2 

II 
u,           «, 

III 
u2            u, 

I 
M2               Da 

II 
«,           «, 

III 

«2               «a 

Ci  and  Ci   St 

10000 
9000 
8000 
7000 
6000 
5000 
4000 
3000 
2000 
1000 
0000 

—3.5      —4.0 
—3.  5      —5.  0 
—3.0      —6.0 
—1.5      —6.5 
0.0      —7.0 
0.0      —7.5 
0.  0      —7.  5 
0.0      —7.0 
-j-1.0      —6.0 
+3.  5      —4.  5 
+4.  0      —2.  5 

+  4.5     —5.5 
+  4.5    —5.0 
+  3.5     —4.5 
0.  0    —4.  5 
-  2.5    —4.5 
+  2.5     —6.0 
+  8.5     —8.0 
+  10.0    —9.5 
+  7.5     —9.5 
+  5.0    —7.5 
+  2.5     —4.0 

+2.5    —  3.0 
+2.5    —  8.0 
+2.0    —  9.0 
—1.5     —  8.0 
—4.0    —  7.5 
—2.5     —  8.0 
0.  0    —10.  0 
+2.0    —12.0 
+2.0    —11.0 
+  1.5     —  7.0 
+1.0          0.0 

—6.5     +3.0 
—3.0     +8.0 
—2.5     +11.0 
—6.5     +13.5 
-9.0     +15.0 
—9.0     +17.0 
—7.  5     +20.  0 
—3.  5     +23.  0 
—1.0     +20.0 
0.0     +8.0 
0.0    +6.0 

—1.0     +  3.5 
—3.0    +11.0 
—5.  5     +13.  5 
—5.0     +  15.0 
—2.0     +15.5 
0.0     +15.5 
0.0     +14.5 
0.0     +13.0 
0.0     +11.0 
—1.5     +8.0 
—3.5     +3.0 

0.0      —3.0 
—1.5      —1.0 
—  2.0      +1.0 
—2.0      +2.0 
—0.5       +3.0 
+  1.5       +5.0 
+2.5      +7.0 
+2.0       +7.5 
0.  0      +5.  0 
—1.5      +4.0 
—2.0      +3.0 

—2.8  -i  :i.-i.  i 

—2.6  +35.0 
—2.4  +34.6 
—2.  2  +30.  0 
—2.  0  +25.  0 
—1.8  +23.6 
—1.6  +22.6 
—1.3  +21.0 
—1.0  +14.0 
—0.8  +6.4 
—0.5  +1.3 

Ci.  Cu  

A.  St  

A.  Cu    

S  Cu 

Cu  and  St 

Wind                     .  .     . 

Means  of  the  velocities,  u.,,  Da  .... 
Radius   o 

—0.  3      —5.  8 
1 
—5.8 

+  4.2    —6.2 
3 
—18.6 

+0.5    •-  7.6 
5 
—38.0 

—4.4    +13.1 
1 
+  13.1 

—2.0    +11.2 
3 
+33.  6 

—0.  3      +3.  8 
5 
+  19.0 

Product   GJV, 

+  MJ  rr  outward  on  radius. 
+  u,  zz  anticlockwise  about  center. 

I.  Oj   =     250,000. 
II.   an  =     750,000. 
III.  om  zz  1,  250,  000. 

+«,  =  south. 

+  D,  zz  oast  . 

TABLE  19.  —  Application  of  the  formula  for  a  cyclone.     (Copy  of  Table  127,  International  Cloud  Report.) 

c     , 

i/)  —  —  -^  tt'Z. 

CONSTANTS  Al 

0  =46°  17'                        "k  — 
u—  —  1.25                      e  = 
CT=  1,250,000  meters 

•JD  FOBMUL^;. 
2ncos0rz.OOO 

-2M       =.000 
a 

fc  =  .000 
A     c 

100 
002 
050 

W  — 

:  +  CZ. 

?..      c. 

=  .000002 

k  —  c  2 
coti 

MZZ  —  -o.                        "-  ' 

1"  k 

-c2oz' 

-k-cz- 

DEBIVED   DISTANCES,    VELOCITIES,    CHECKS,    AND    INCLINATIONS. 

0 

mftfrt. 
1250000 

„,  ''/<  .-•. 

777 

z 
1.00 

OS 

1250000 

u 
—1.25 

V 

2.50 

w 
.0000020 

av 
3125000 

j   otozz  I 
\-1uz\ 

2.5 

cot; 

2.00 

i 
26.6 

O 

„,„,. 
0.34 

1000000 

621 

1.56 

1560000 

—1.0 

3.125 

31 

3125000 

3.1 

3.12 

17.6 

0.  40 

750000 

466 

2.78 

2085000 

—0.8 

4.  17 

56 

3125000 

4.2 

5.56 

10.2 

0.55 

500000 

311 

6.25 

3125000 

—0.5 

6.25 

125 

3125000 

6.3 

12.  50 

4.6 

0.78 

250000 

155 

25.00 

6250000 

—0.25 

12.50 

500 

3125000 

12.5 

50.00 

1.2 

2.03 

200000 

124 

39.06 

7812000 

—0.20 

15.  61 

781 

3125000 

15.6 

78.12 

0.7 

3.00 

150000 

93 

69.40 

10412000 

—0.15 

20.82 

1388 

3125000 

20.8 

138.  80 

0.4 

5.  37 

100000 

62 

156.  30 

15630000 

—0.10 

31.26 

3126 

3125000 

31.3 

312.  GO 

0.2 

13.90 

50000 

31 

625.00 

31250000 

—0.05 

62.  50 

12500 

3125000 

62.5 

40000 

25 

977.00 

39080000 

—0.04 

78.  16 

19540 

3125000 

78.2 

30000 

19 

1736.  00 

52080000 

—0.03 

104.16 

34720 

3125000 

104.2 

20000 

12 

3906.  00 

78120000 

—0.02 

156.24 

78120 

3125000 

150.  2 

10000 

6 

15625.  00 

156250000 

—0.  005 

312.  50 

.  0312500 

3125000 

312.5 

43 


was  shown  by  the  results  of  the  observations  that  the  move- 
ment at  the  cumulus  level  is  much  more  rounded  than  in  the 
lower  strata,  the  difference  being  caused  by  the  retardation  of 
the  air  operating  upon  the  surface  irregularities  of  the  ground. 
A  congested  or  irregular  inflow  near  the  center  will  similarly 
increase  the  angle  i,  since  the  component  u  is  increased  and  v 
diminished  by  it.  The  observations  given  on  the  weather 
maps  do  not  record  the  conditions  within  60  miles  of  the 
center  with  any  definiteness.  In  hurricanes  the  core  is  about 
30  miles  broad  and  its  boundary  is  quite  sharp,  which  shows 
that  the  component  v  is  highly  developed,  while  u  is  small. 
But  the  hurricane  also  penetrates  to  much  greater  altitudes, 
as  already  mentioned.  Finally,  the  gradient  shows  that  there 
is  a  slow  change  near  the  outer  limit,  but  that  it  increases 
very  rapidly  on  approaching  close  to  the  center. 

THE  SPECIAL    FEATURES    OF    THE  CIRCULATION. 

The  special  features  of  the  circulation  indicated  on  fig.  20, 
which  may  be  mentioned  are  as  follows:  (1)  It  is  evident  that 


(3)  The  approach  of  a  moving  particle  to  the  axis  of  the  cyclone 
is  attended  by  an  increase  of  the  velocity  of  rotation,  which  ac- 
celerates rapidly  as  it  passes  into  the  upper  strata.  There  it 
accomplishes  the  work  of  deflecting  the  eastward  drift,  and  it 
expends  some  of  its  energy  in  that  way.  The  result  of  this  op- 
position to  free  motion  is  to  spread  out  the  top  of  the  vortex, 
reduce  its  gyratory  velocity,  and  change  the  relations  of  i>2  and 
ut.  In  the  undisturbed  gyratory  motion  the  component  c2  be- 
comes very  great  in  comparison  with  w2,  and  «2  is  always  a 
small  quantity.  An  inspection  of  Table  18  shows,  however, 
that  in  the  strata,  between  5,000  and  10,000  meters  the  radial 
velocity  U2  is  relatively  large,  and  the  angle  of  the  inclination 
instead  of  being  nearly  0°  is  from  25°  to  35°  on  the  inner 
circle  II.  This  means  that  the  original  coefficient  upon  which 

2w 
the  dimensions  of  the  cyclone  depend,  namely,  c=  — — ,  does 

not  remain  a  constant,  but  increases  from  the  boundary  to- 
ward the  axis  of  the  gyration.  Thus  we  obtain, 


.s      / 


/     /     / 


/     / 


s*     ^r 


20,000  m. 
S 

<9 
7 
6 


lOOOkm. 


FIG.  20. — General  scheme  of  the  structure  of  cyclones. 


this  scheme  avoids  entirely  the  primary  difficulties  attending 
the  Ferrel  and  also  the  German  types  of  circulation.  Each  of 
these  divided  the  cyclone  into  two  parts  having  special  prop- 
erties. Ferrel  divided  his  cyclone  at  a  vanishing  rotational 
velocity,  v  =  0,  which  involved  a  circulation  in  opposite  direc- 
tions on  either  side  of  it;  the  German  type  consists  of  two 
parts,  separated  by  a  discontinuous  movement  at  the  circle  of 
the  maximum  velocity  for  v.  The  pure  vortex,  on  the  other 
hand,  has  only  one  law  to  deal  with,  and  that,  too,  the  simplest 
of  all,  in  accordance  with  which  the  motion  is  generated.  (2) 
Neither  of  the  other  types  provides  for  a  true  calm  region  at 
the  center  of  the  cyclone,  commonly  observed  in  hurricanes 
as  the  eye  of  the  storm.  Ferrel's  formula  402i  shows  that  v 
increases  from  the  circle  J?  =  0.707  to  the  very  center;  the 
formula;  450  and  471  show  that  the  velocity  v  decreases  grad- 
ually from  the  circle  bounding  the  inner  region  toward  the 
center,  but  does  not  vanish  till  reaching  it.  The  construction 
here  proposed  indicates  that  since  u  is  a  function  of  the  height 
z  as  well  as  of  the  radial  distance  ro,  the  air  in  streaming  to- 
ward  the  center  is  gradiially  lifted  above  the  ground  by  purely 
dynamic  action  and  leaves  a  core  without  gyratory  circulation. 


CTI  =  .0000053= 


0.6 

1250000 
4.0 


eTTT  =.0000005= 


8.8 
250000 


This  involves  an  expenditure  of  energy  in  the  struggle 
attendant  upon  intruding  into  the  swiftly  moving  upper 
stratum.  Furthermore,  as  was  previously  pointed  out,  Fer- 
rel's theory  of  the  slowing  down  of  the  excessive  eastward 
velocities  which  would  arise  from  the  pure  vortex  law  of  the 
conservation  of  areas  applied  to  the  general  cyclone,  is  ths, 
friction  is  largely  concerned  with  the  operation.  It  seemis 
however,  that  a  much  more  efficient  cause  of  retardation  at 
the  interaction  between  these  two  types  of  motion,  namely, 
the  linear  and  the  rotary,  by  which  the  lower  strata  thrust 
themselves  into  the  higher.  The  effect  is  to  enlarge  the  size 
of  the  vortex  tube  at  the  top  by  the  resistance,  deflect  the 
eastward  drift  into  sinuous  curves,  slow  down  the  eastward 
velocity,  and  thus  restrain  the  general  cyclonic  movement  from 


TAIU.K  2(1. 


44 

uiul  ci'loritieft  in  the  waterspout  off  Cottage  City,  Viiu'i/anl  Sound.  Mti-xx.,  Aiujunt  /!>,  /,"?.%'.     (Copy  of  Table  I2S  International  Cloud 

Iti-port.  I 


Working  formula! :        «2  — 2o«;        «=     ;       ro  =  "  ;       or  — Const.;       2zit=taw; 

3  to  £         VGJ 


Dimensions  in  meters. 

Velocities  ill  meters  per  second. 

iM'mcn 

-ion-  in 

feel. 

Velocity  in  miles  JMT  hour. 

ft             Z 

a 

tt 

V 

w 

2ZM  ~  GJU) 

ft 

Z 

a 

H 

V 

w 

1280          0 

00 



0 

0 

0 

4200 

0 

00 



e 

0 

1278          2 

518.4 

3.13 

6.26 

0.02 

12.53 

4193 

7 

1701 

7.0 

11.  i 

0.04 

1189        91 

76.9 

0.46 

42.24 

1.  10 

84.49 

3901 

299 

253 

1.0 

B4  I 

2.5 

1097       183 

GO.  8 

0.29 

53.39 

1.76 

106.  79 

3599 

601 

200 

0.  6 

111).  5 

3.9 

1006      274 

44.3 

0.27 

73.31 

3.31 

146.  61 

3301 

899 

145 

0.6 

164. 

7.4 

914      366 

38.3 

0.23 

84.72 

4.42 

169.  45 

2999 

1201 

125 

0.5 

189. 

1).  D 

731       549 

31.3 

0.19 

103.  77 

6.63 

207.  53 

2398 

1802 

102 

0.4 

233. 

11.'.) 

549      731 

27.1 

0.16 

119.  73 

8.83 

239.  47 

1802 

2308 

89 

0.4 

268. 

19.8 

457      823 

25.5 

0.15 

127.04 

9.94 

254.  10 

1409 

2701 

84 

0.3 

284. 

22.2 

366      914 

24.3 

0.15 

133.  89 

11.04 

267.78 

1201 

2999 

79 

0.3 

300. 

24.7 

183     1097 

22.1 

0.13 

146.  68 

13.25 

293.  36 

601 

3599 

72 

0.3 

328. 

29.  G 

146     1134 

21.8 

0.13 

149.  13 

13.70 

298.'  26 

479 

3721 

72 

0.3 

333. 

29.7 

0     1280 

20.5 

0.13 

158.  44 

15.46 

316.  89 

0 

4200 

67 

0.3 

354. 

34.6 

=  0.01208 


excessive  values.  (4)  The  resultant  of  these  component  forces 
and  velocities  is  to  produce  a  circulation  along  the  parallels 
of  latitude  which  may  be  represented  by  the  upper  part  of  fig. 
20.  The  upper  clouds  of  the  cirrus  region  precede  the  cyclone 
proper  as  forerunners  of  this  type  of  circulation;  the  lower 
clouds  follow  in  succession,  till  precipitation  is  produced  at 
the  elevation  of  1,000  to  3,000  meters;  in  very  rapid  circulations 
the  eye  of  the  storm  is  fully  developed;  the  clearing  up  is 
more  abrupt  on  the  westward  than  on  the  eastward  side  of  the 
cyclone.  (5)  The  progressive  movement  of  storms  is  partly 
an  effect  of  the  cyclone  covering  an  area  of  sufficient  extent 
to  be  in  different  latitudes,  so  that  variations  in  cos  0  amount 
to  something.  But  other  circumstances  are  more  important. 
By  Table  33,  Section  IV,  International  Cloud  Report,  it  is  seen 
that  the  northward  components  for  the  group  of  areas  which 
are  covered  by  the  currents  of  air  from  the  south  are  greater 
than  those  from  the  north.  That  is  to  say,  the  movement  in 
the  streams  from  the  Tropics  is  more  rapid  than  that  from 
the  polar  regions,  and  the  result  is  to  roll  up  the  eastward 
side  of  the  cyclone  to  the  north  more  than  the  westward  to 
the  south.  The  cyclone  tends  to  rotate  along  the  front  of  a 
high  area  toward  the  north.  At  the  same  time  its  top  is 
fastened  by  means  of  the  circulation  into  the  eastward  drift  at 
4,000  meters  elevation,  and  these  two  components  make  the 
storm  move  northeastward  in  the  central  and  eastern  portions 
of  the  United  States.  If  the  general  cyclone  has  other  direc- 
tions, as  when  hurricanes  form  in  the  Caribbean  Sea,  the  same 
principles  hold,  and  the  storm  there  first  moves  westward,  then 
northward  and  northeastward,  because  the  general  circulation 
is  controlled  in  that  region  by  the  anticyclone  in  the  southern 
portions  of  the  north  Atlantic  Ocean.  (6)  It  has  been  shown 
that  the  currents  which  feed  the  cyclone  have  different  veloci- 
ties at  different  altitudes,  being  greatest  from  2,000  to  5,000  me- 
ters above  the  ground.  Each  stratum  forms  a  stream  for  itself 
conforming  to  the  general  law,  but  modifies  its  dimensions 
according  to  the  constants  pertaining  to  the  special  locality. 
Since  these  different  strata  have  thus  distinct  local  movements, 
especially  considering  the  variable  temperatures  and  densities 
of  the  currents  from  the  north  and  south,  respectively,  it  fol- 


lows that  the  conditions  are  favorable  for  the  formation  of  tur- 
bulent minor  circulations  of  all  kinds.  The  movement  of  the 
air  is  therefore  partly  congested,  and  partly  runs  in  free  whirls, 
the  difference  of  equilibrium  in  temperature  being  gradually 
reduced  to  the  proper  normal  value  for  the  latitude  anil  alti- 
tude by  this  forced  intermingling  of  the  subordinate  parts  of 
the  cyclone.  This  process  of  restoring  to  an  equilibrium  the 
temperature  of  masses,  bearing  with  them  that  of  the  region 
from  which  they  came,  is  generally  completed  by  the  time  the 
5,000-meter  level  is  reached,  judging  from  the  records  of  the 
balloon  observations.  In  summer,  when  the  eastward  drift  is 
relatively  slow,  the  pure  vertical  convective  ascension  may  ex- 
tend up  to  10,000  meters.  This  is  much  more  likely  to  happen 
on  the  eastward  than  on  the  westward  side  of  a  cyclone,  be- 
cause the  vortex  components  throw  back  the  eastward  move- 
ment upon  itself,  and  thus  make  the  strata  more  stagnant  in 
vertical  directions.  It  has  also  been  found  practically  very 
difficult  to  make  the  kites  fly  on  the  east  of  the  low  center,  the 
best  ascensions  being  made  on  the  southerly  and  westerly 
quadrants.  (7)  -The  entire  problem  of  amily/.ing  the  move- 
ments of  the  air  in  their  details  is  so  exceedingly  complicated 
that  only  slow  improvements  in  dynamic  meteorology  can  be 
expected.  A  clearer  idea  of  the  fundamental  conditions  may, 
however,  enable  us  to  advance  more  rapidly  than  is  now  antic- 
ipated. It  will  be  a  very  important  gain  if  meteorology  can 
free  itself  from  some  of  the  theories  which  have  so  long  pre- 
vailed, but  which  now  are  seen  to  be  quite  untenable,  and  have 
seriously  retarded  its  advancement. 

THE    VELOCITIES    IN    TORNADOES. 

The  motions  in  tornadoes  are  similar  to  those  in  cyclones, 
yet  the  tube  is  not  only  inverted  in  position,  but  the  stream 
lines  occupy  only  the  central  portions,  and  the  lines  of  >,'•  become 
tangent  to  the  plane  whose  height  is  7/o  at  certain  distances  from 
the  center.  The  fundamental  equations  for  the  tornado  are 

303-308.      <,'•  •§-(-—  ro'g  holds  for  the  tornado  with  vertical  axis 

4J 

positive  downward,  and  the  single  bounding  plane  at  the  dis- 


45 


tance  H0  above  the  ground.     4'  =  — o  w*s  ™  ^e  equation  for 

the  cyclone  with  the  vertical  axis  positive  upward.  A  multitude 
of  minor  relations  and  comparisons  can  be  drawn  from  the  two 
sets  of  equations  303-308  and  488-490.  If  the  lower  parts  of 
fig.  20  be  looked  at  as  if  the  horizontal  axis  were  the  vertical, 
we  have  a  picture  of  the  half  of  a  tornado  tube.  The  diagram 
is  not  good  because  not  drawn  to  scale,  but  the  idea  is  easily 
understood.  Hence,  one  law  serves  for  all  types  of  local 
storms,  cyclones,  hurricanes,  and  tornadoes,  which  thus  theo- 
retically differ  from  each  other  only  in  their  dimensions  and 
in  the  details  by  which  they  are  formed.  Cyclones  are  gener- 
ated chiefly  by  horizontal  convection  currents;  hurricanes 
have  a  stronger  vertical  convection  current  and  also  horizontal 
convection  currents ;  tornadoes  arise  chiefly  from  vertical  con- 


vection, assisted  by  some   horizontal  currents  which  counter 
flow  in  the  cumulus  level. 

It  may  be  remarked  that  the  stream  lines  indicated  by  Fer- 
rel,  page  300,  Recent  Advances,  are  conjectural  only,  and  do 
not  conform  to  the  theory  of  stream  lines  in  a  vertical  vortex 
tube,  nor  to  observation,  which  shows  that  the  air  is  quiet  close 
up  to  the  boundary  of  the  tornado  tube. 

THE    WATERSPOUT    OFF    COTTAGE    CITY,   MASS.,  AUGUST    19,   1896. 

The  result  of  the  computation  on  this  interesting  waterspout 
is  added,  and  it  shows  the  dimensions  and  velocities  in  metric 
and  English  measures  which  were  derived  from  the  observed 
distances  and  the  formulae.  The  most  important  feature  is  the 
value  of  the  vertical  velocity  of  35  miles  per  hour  at  the  sea 
level.  See  Table  20. 


VI.— CERTAIN  MATHEMATICAL  FORMULAE  USEFUL  IN  METEOROLOGICAL  DISCUSSIONS.' 


THE  NEED  OF  A  STANDARD  SYSTEM  OF    FOEML'L*. 

There  are  a  large  number  of  mathematical  papers  that  have 
been  written  by  meteorologists  ill  the  exposition  of  various 
theories,  which  must  be  thoroughly  considered  by  students  whp 
seek  to  go  beyond  a  descriptive  statement  of  the  problems 
into  a  close  examination  of  the  principles  upon  which  the 
solutions  rest.  The  question  arose  at  an  early  stage  in  my 
study  of  comparative  meteorology  as  to  the  form  in  which 
such  mathematical  discussions  would  be  presented  to  the 
public.  To  traverse  the  entire  range  of  treatises  and  explain 
them  in  detail  was  clearly  impracticable;  to  adopt  an  abstract 
mathematical  synopsis,  such  as  is  found  in  Carr's  or  Laska's 
synopsis  of  pure  mathematics,  was  to  put  too  great  a  strain 
upon  readers  who  are  not  specialists  in  mathematical  meteor- 
ology. Finally  it  seemed  to  me  to  be  a  fair  compromise  to 
take  the  following  course:  (1)  reduce  the  important  papers,  to 
one  common  standard  notation,  and  (2)  make  an  analysis  of 
the  result  in  a  sufficiently  expanded  form  to  enable  a  good 
reader  to  follow  the  series  of  equations  without  difficulty. 
The  only  step  required  to  transform  the  contents  of  the  mathe- 
matical compendium  as  given  in  chapters  10  and  11  of  the 
International  Cloud  Keport  into  a  complete  treatise  on  ana- 
lytic meteorology  is  to  supply  such  transition  precepts  as  are 
usually  placed  between  the  formulre  to  aid  the  thought.  It 
is,  however,  a  distinct  advantage  for  a  working  use  of  the  for- 
mula, to  one  who  has  once  become  familiar  with  such  prob- 
lems, to  dispense  with  these  explanatory  sentences,  which 
only  take  up  space.  A  ready  reference  to  the  standard  equa- 
tions under  each  subject  is  quickly  appreciated  by  any  one 
who  uses  these  formulae  in  a  practical  way,  just  as  one  would 
use  a  mathematical  table  in  computing.  It  is  my  purpose  to 
complete  such  a  collection  of  formulie,  in  addition  to  the 
tables  contained  in  my  report  on  Eclipse  Meteorology  and 
Allied  Problems,  Weather  Bureau  Bulletin  I,  1902,  by  appro- 
priate tables  covering  the  subjects,  spherical  harmonics,  ther- 
modynamics, and  the  kinetic  theory  of  gases,  because  these 
are  indispensable  in  meteorological  studies.  I  have  taken  the 
opportunity  in  this  connection  to  present  several  original  sets 
of  formula),  which  have  an  advantage  in  their  applications  to 
meteorological  problems,  and  it  is  my  purpose  to  call  atten- 
tion to  some  of  them  in  this  paper. 

THE  GENERAL  EQUATIONS  OF  MOTION. 

The  methods  of  deriving  the  general  equations  of  motion 
on  the  rotating  earth,  as  presented  in  Ferrel's  paper,  "The  mo- 
tions of  fluids  and  solids  on  the  earth's  surface,"  or  in  the 
standard  treatises  of  hydrodynamics,  are  so  complicated  as  to 
discourage  all  who  are  not  expert  mathematicians  from  an  ex- 
amination of  the  solution.  The  fact  that  Ferrel  did  not 

evaluate  the  total  differential  of  inertia  -  (^^"l,  introduced 

at 

an  error  into  the  equations  contained  in  his  "Mechanics  and 
general  motions  of  the  atmosphere,"  United  States  Coast 
Survey  lleport,  1875,  Appendix  20;  this  was  eliminated  in  his 
"  Recent  advances  in  meteorology,"  Annual  lleport  of  the  Chief 
Signal  Officer,  1885,  Appendix  71.  There  are  no  doubt  many 
ways  of  solving  this  problem,  but  the  following  is  original, 
as  expanded  from  Table  75,  International  Cloud  Report,  and 
it  leaves  little  to  be  desired  in  respect  of  simplicity  and 
completeness. 

(1)  THE  POLAR  EQUATIONS  OF  MOTION  ON  THE  ROTATING  EARTH. 

Using   the  notation   already   adopted  in  Paper  II  of   this 


1  Keprinted  from  the  Monthly  Weather  Review  for  June,  1902. 


series,2  we  write  the  primary  equations  of  acceleration  of 
motion  referred  to  axes  which  have  their  origin  at  the  center 
of  a  nonrotating  earth,  as  follows: 

The  accelerations  due  to  motion  and  to  external  forces  are, 


155. 


I  OP      dV      du 

i      =  "jT  —  Voti  +  w<a, 

dx        at 


"5 

p  dx 
IdP 


dV      do 


1  <9P      dV      dw 
~~dz~ 


where  the  angular  velocities  of  motion  for  a  point  are 
166.  v  .    u  .        v 

(a,  =  —  — 
i  ,. 


i     u 

<",=     +- 

r 


"  r  tan  6 

Compare  diagram  in  my  Report,  page  498,  or  Basset,  pages 
13  and  14,  noting  the  transformations  of  notation. 

In  case  the  earth  rotates  with  the  constant  angular  velocity  n, 
carrying  the  fixed  axes  with  it,  the  linear  velocities  (u,  v,  w) 
and  the  angular  velocities  (««,,  u>3,  tus)  are  changed  as  follows, 
denoting  these  terms  on  the  rotating  earth  with  primes: 


177.  u'  =  u 

v'  =  v  -f  n  r  sin  0 
w'  =  w 


178. 


v  +  n  r  sin  0 


u 
r 


i        i 


v  +  n  r  sin  0 


The   differentials 


179. 


d  (n  r  sin  0)       du 


r  tan  0 

This  is  due  to  the  fact  that  the  rotation  of  the  earth  adds 
the  velocity  n  r  sin  0  =  n  w  to  the  eastward  linear  velocity, 
because  m  is  the  perpendicular  distance  from  the  axis  of 
rotation. 

du'     dv'     dw' 

-JT->    jr>    ~JT~  evaluate  into, 

dt       dt      dt 

du'       du 
dt    '~  dt 

dv'  dv 

dt  dt 

dw'  dw 

dt  dt 

since  M  =  !^?  and  w=  —  by  formulae  153,  page  497,  of  the  In- 
dt  dt 

ternational  Cloud  Report. 

Substituting  these  values  in  the  equations  of  motion  for  the 
rotating  earth,  which  are  the  same  as  those  of  155  with  the 
letters  all  primed,  and  taking  the  equivalents  of  dx,  dy,  dz  in 
polar  cordinates  from  153,  we  have: 


dt 


=  -=T  +  u  n  cos  0  -f  w  n  sin  0 
dt 


180. 


IdP 

"prdO 

1  _  d-F 
'  P 

1^9P 
Or 


du  .        (v  +  nrnhi/i) 

=  dt_(v  +  nrmn0)—T-^0- 


+ 


u 

U  — : 

r 


dv 


(v  +  n  r  sin  0)          (v+nr  Bind) 


r  tan  0 


+  un  cos  0  +  w  n  sin  0, 


dw 


(v-\-nrsmO) 


The  external  forces  derived  from  the  potential  Fare: 

dV  dV  1T=_ 

~  dx  ~'    '        ~  dy  ~  dz 


2  See  Monthly  Weather  Eevlew  for  February,  1902,  Vol.  XXX,  p.  81. 

47 


48 


Performing  the  algebraic  work,  these  equations  reduce  to 


181. 


dP 


,11' 


ilv 
=  ,„  + 


p  r  sin  (Id).      (It 


1  ill' 


I'2  COt  l>  -f-  /HI' 

r 

2/i  cos  0  .  v  —  r  «s 
MI-  cot  0  -\-  vw 

r 
2»  cos  (I 


sin  U  cos  0  , 


w*  -f 


a  +  "2n 
—  2/i  sin  ti 


v  —  rri'  sin2  0. 


The  successive  terms  are  the  inertia,  the  centrifugal  forces, 
the  deflecting  force,  and  the  forces  which  change  the  figure  of 
the  earth  from  a  sphere  into  an  ellipsoid  of  revolution. 

(2)  THE  CYLINDRICAL  EQUATIONS  OF  MOTION  ON  THE  ROTATING 

EAKTH. 

If  the  axis  of  rotation  of  the  earth  is  taken  as  the  axis  of 
rotation  in  cylindrical  coordinates,  the  tangential  velocity 
_  y  _|_  n  w;  but  if  the  axis  of  rotation  is  any  radius  of  the 
earth  extended  above  the  surface,  the  tangential  velocity  be- 
comes =  o  -f-  n  w  cos  0.  Hence  we  have,  in  cylindrical  coor- 
dinates, 
182.  u'  =M 

v '  =  v  +  n  ra  cos  U 


w\  =  0 


w 


w 


,  =  0 

, =  n  cos  0 


V 

w 


The  differentials  —  —  ,    , 
at      at 


-  — 
at 


evaluate  into, 


183. 


du' 
"rfT 


du 
dt 


dv'       du   .   d  (n  m  cos  0)       do   . 

= —  -f-  ..._^..=       +  u  n  cos  0 

dt        dt  dt  dt 

dw ' dw 

dt      "dt 

since  u  =       ,  by  formula;  152,  and  cos  0  is  a  constant. 


Sub- 


stituting these  values  in  the  equations  of  motion  for  the  rota- 
ting earth,  which  are  the  same  as  those  of  155,  with  the  letters 
all  primed,  and  taking  the  equivalents  of  dx,  dy,  dz,  in  cylin- 
drical coordinates  from  152,  we  have: 
184.  1  dP  du 


1    dP 


dv  f  v\ 

=  -r.  +  u  n  cos  6  +  M  (  n  cos  0  -| —  I 


1  OP 


dw 
~dt' 


The  external  forces  derived  from  the  potential  Fare: 

dV  dV  dV 

=0,  —      =0, _  —  a. 

dx  dy  dz 

Performing  the  algebraic  work,  these  equations  reduce  to 


185. 


du  v'' 

=  -JT  —  2n  cos  0 .  v 

at  zzj 


du 


—  (2n  cos  0  +  i/j)  v 


I  dP 


dv 


uv 


,.  +  2n  cos  0.  u  .  + 


1  HI' 
7  <5r 


=    "  +  (2n  cos  0  +  v,) 
dt 

dw 
=  ~dt 


vhere   the  term  +  ra  n*  cos*  0  is  neglected  in  the  first  equn- 

,ion  and  v  =  —  the  relative  angular  velocity. 
1       ro 

The  successive  terms  are  the  inertia,  the  deflecting  force, 
ind  the  centrifugal  forces, 

REMARKS     ON     THE     SEVERAL     TERMS     IX     THE     GENERAL     EQUATIONS    (IF 

MOTION. 

It  is  customary  to  add  to  the  terms  developed  in  a  friction- 
ess  medium,  a  term  expressing  the  retardation  of  acceleration 
due  to  friction,  either  in  Ferrel's  form  +  k  (u,  v,  »•),  which  is 
oroportional  to  the  velocity  and  expresses  a  sliding  friction, 

or  in  Oberbeck's  form  —  <d*  (u,  v,  w),  which  expresses  a  retarda- 
tion proportional  to  the  turbulent  internal  resistances  of  a 
mixing  fluid.  This  function  is  hard  to  evaluate  on  account 
of  the  uncertainty  which  attaches  to  the  invisible  internal  mo- 
tions, and  to  the  effect  of  discontinuous  surfaces  separating  dif- 
ferent velocities  and  temperatures.  Near  the  ground  turbulent 
motions  and  large  coefficients  of  friction  up  to  about  300-500 
meters  are  required ;  above  this  level  and  especially  in  the  higher 
strata  the  coefficient  of  friction  is  very  small. 

The  inertia  terms  -    ~3r'~      disappear  in  steady  motion,  and 

they  are  small  in  slow  changes  of  velocities.  There  are,  how- 
ever, cases  in  which  inertia  may  amount  to  a  considerable 
quantity,  as  where  a  tornado,  in  passing  along  its  path,  sucks 
in  new  masses  of  air,  and  transforms  them  suddenly  from  rest 
into  excessively  rapid  motion.  Also,  when  the  cyclonic  vortex 
raises  masses  of  air  from  strata  having  slow  motion  into  strata 
of  rapid  velocities;  but  especially  where  countercurrents  meet, 
and  the  stream  lines  are  bent  and  reflexed  in  their  direction. 

These  two  terms,  friction  and  inertia,  act  in  the  path  of  mo- 
tion and  they  directly  affect  the  quantity  of  kinetic  energy 
possessed  by  the  elementary  masses.  All  forces  which  act  at 
right  angles  to  the  path,  such  as  the  centrifugal  and  the  de- 
flecting forces,  do  not  change  the  momentum,  but  they  do  alter 
the  direction  of  the  path.  Hence,  in  integrating  for  the  kinetic 
energy  in  an  orbit,  or  in  a  circuit,  the  centrifugal  and  the  de- 
flecting forces  drop  out  of  the  equations,  but  they  must  be 
retained  when  discussing  the  angle  that  the  stream  line  makes 
with  the  isobars,  which  angle  expresses  the  influence  of  the 
velocity  potential  function  on  the  motion.  The  following  inte- 
gration of  the  general  equations  will  establish  these  proposi- 
tions. 

INTEGRATION      OF      THE     GENERAL     EQUATIONS      OF      MOTION      IN     1'nl.U; 

COORDINATES. 

Make  the  following  substitutions  in  181: 
197. 


-  =  v  sin  0 
r 


v1  cot  0 

=  V  COS  0  .  V 

r 

u  v  cot  0 

=  u  cos  0  .  v 
r 

and  neglect  the  terms  in   w2,  which   are  very  small,  with   the 

result  that, 

200.  1  dP       du  UK- 


—  —  _  _  _ '  +  cos  0  (2»  +  •/)  //  +  sin  n  (2«  +  *)  «• 
/>  dy        dt 

1  dP      dw         .  u' 

Now  multiply  these  equations  respectively  by  </./-.  '/</,  <ls,  and 
remember  that  rdx  =  udy,  wft.r  =  iu'):,  u'dy  =  n)s;  take  the  sum 
of  the  partial  differentials,  the  result  being, 


49 


f)(\f>  op 

-  =  udu  +  i-Ov  +  wdw  +  gdz. 

The  integral  of  this  is, 

/(>/' 
—  —  =  i  («J  +  u2  +  w2)  +  yz  +  const.  =  £  <?2  -f  j/s  -f  const. 
/' 

This  is  the  fundamental  equation  of  steady  motion  found  in 
all  treatises  on  hydrodynamics;  its  discussion  is  carried  on  in 
Table  81,  International  Cloud  Report.  The  centrifugal  and 
the  deflecting  forces  have  disappeared,  and  the  integral  is 
equivalent  to  the  kinetic  energy,  |  </*,  plus  the  external  force  due 
to  the  acceleration  of  gravitation.  An  arbitrary  term  may  be 
added  to  express  the  frictioual  retardation. 

If.  the  integration  is  between  two  points  of  a  fluid  that  has  the 

C       OP  P 

same   density   throughout,   the    term     I —  =  —  —  simply. 

Such  lines  of  homogeneous  integration  may  be  found  by  ob- 
serving the  surfaces  of  equal  density  in  the  atmosphere,  or,  a 
mean  density  between  two  points  may  be  assumed  in  place  of 
the  existing  variable  density.  If  the  velocity  term  ^  q*  is  neg- 
lected, we  obtain  dP  =  —  g/'dz,  and  this  is  the  simple  form 
from  which  the  usual  hypsometric  formulae  for  barometric  re- 
ductions are  derived.  Compare  formulae  54,  Table  66,  p.  490. 

It  is  noted,  however,  that  the  usual  method  employed  in 
static  barometric  reductions  is  incomplete,  and  that  the  ve- 
locity term  J  (if  —  q*),  where  q,  qa  are  the  observed  velocities 
at  the  two  points  limiting  the  path  of  integration,  has  been 
omitted. 

If  the  integration  is  continued  in  any  closed  circuit  the 
gravity  term  disappears  from  the  equation,  and  the  velocity 
terms  alone  remain.  This  line  integral  (C.)  measures  the  work 
done  in  moving  the  unit  mass  once  around  the  circuit,  while 

A  ='   '  is  the  rate  of  doing  the  work,  or  the  activity.     From 

this  point  of  view  the  circulation  of  the  atmosphere  may  be 
treated  by  the  ordinary  theory  of  the  line  integral.  It  is  more 
convenient  to  observe  the  velocities  than  the  pressures  and 
densities  around  a  circuit,  in  the  present  state  of  meteorology. 

EXPRESSIONS   FOB    THE    GRADIENTS    OF    PRESSURE. 

If  we  take  the  formulae  for  acceleration,  Cloud  Report,  page 
499, 


155. 


dP  _dV 

()  '/*          ()  Ti 

dP       dV 


dz 


- 

~0z 


we  can  write  for  the  gradient, 
501. 


n  dP  dV  . 

G    =  —  --  —  p  —  =  i>  u. 
dx       '  dx 

G          _  OP  _       dV  =       ?. 
y  "          d  ii        ''  d<l         P     ' 


dP 


o,  =  -      -p~  =  MV 

dz  dz 

dV  dV 

The  gradient  terms,  —  ti  —  in  latitude,  and  —  p  —  -   in  lon- 
'   dx  '  dy 

d  V 
gitude,  are  small  terms,  while  —  p—-  is    the    principal    term, 

dz 


3  The  series  of  equations  beginning  with  501  may  be  considered  as  an 
cxlriisioii  of  the  system  given  in  the  International  Cloud  Report,  which 
ends  on  page  G03. 


and   these   are   due   to  the  attraction  of  the  earth  upon  the 

dP  dP 

atmosphere.     The  terms  —  —  in  latitude, in  longitude, 

dx  dy 

dP 

and  —          in  altitude  are  the   gradients  due  to  the  thermal 
dz 

listurbauce  of  the  isobaric  surfaces,  the  first  two  being  the 
gradient  terms  producing  the  horizontal  flow  of  the  atmos- 
phere, and  the  last  one  the  term  which  causes  the  up  and 
down  movement  of  the  atmosphere  by  the  variations  of  the 
normal  buoyancy  from  that  of  stable  equilibrium  as  controlled 
by  the  static  terms  in  the  potential  function  for  external  force  F. 

It  is  next  important  to  evaluate  the  gradient  terms  for  use 
in  practical  meteorology.  There  are  many  ways  of  doing  this, 
as  is  indicated  by  the  collection  of  formulae  in  Table  65,  page 
489,  of  the  International  Cloud  Report.  There  is  a  generally 
accepted  convention  which  is  adopted  as  the  basis  for  the 
practical  measures  of  gradients  by  the  mercurial  barometer. 

Thus,  the  difference  of  barometric  pressure,  G,  at  two  points 
which  are  111  111  meters  apart  in  a  horizontal  direction,  is 
taken  as  the  standard  for  reductions. 


B—j — ^= ^^H 

ctx  ^ BQ 

Fia.  21. — Vertical  section  through  the  atmosphere. 

In  fig.  21,  which  shows  a  vertical  section  through  the  atmos- 
phere, let 

D  =  111  111  meters  =  1°  on  surface  of  the  earth, 
dx  =  1  meter, 

El  =  any  distance  between  given  points  of  observation. 
Then, 


502. 


G 


D 


D       111  111 

,  D 


AE  . 

~dx  ' 


It  is  necessary  first  to  find  G  from  the  observed  values  of 
B  at  two  stations  at  the  distance  A\  from  each  other. 

dh 

EVALUATION  OF  THE  COEFFICIENT   ,„  AND  OTHER  TERMS. 

If  the  change  in  elevation  of  the  isobaric  surface  is  as  follows: 
ht  at  distance  Ev  h0  at  distance  D,  dh  at  distance  dx,  then, 

503.  h,          h0         dh 

g   i  =  g^-  =  gr-r-  =  g  tan  «  =  go  measures  the  acceleration. 
j^j  J.S         dx 

\ 

Also  by  the  law  of  falling  bodies,  v  =  ^/2gh  for  the  velocity, 
I.   We  have, 

504.  dh      h,,        h.         I    , 

—  =  -^  =   J-  =  „  for  the  top  of  the  homogeneous  at- 
dx      D       jft/j        .&, 

mosphere. 

505.  dli_B,—B__G 

D  = 


dx~       D 
of  the  Cloud  Report. 

Hence, 
506.    dh  h,,  I 


I) 

6 


=       .          ,  from  24,  p.  487 

dx    ,/„,,„,' 


El  I 


RT 


_     _ 

~  E,  I^-Ji  -  Et  1^— 

Et  E,          E, 


50 


because  B  at  the  .  top  of  the  homogeneous  column  I  is  negli- 
gible compared  with  Bt  at  the  bottom  of  it.  B,  is  in  this  con- 
nection the  barometric  pressure  at  the  surface,  and  Ji,  =  />'„ 
=  0.760  meter. 

II.   We  have  by  50,  page  490,  for  the  standard  weight  of  the 
atmosphere, 

507.  p0  =  «h  =  <rj  =  amHn  .     Hence, 


Bn.     That  is,  h  =  I  for  a  =  an.     Hence, 


509. 
510. 

511. 


=  l  =  ^Bn.     Therefore, 

a 


_      _    _  _ 

8  ~  T:  —  K  —  T 


RT  '  —  13»595-8  _  7,991.04 
'  '  1.29305 


B. 


0.760 
=  10,514.5  dB  =  s  dB. 


10,514.5. 


512.  dh  =  10  514  5  BO  —  B_  =  10,514.5 -  _  =  0.09463(7. 

dx  D  111  111 

III.  Let  /'  =  the  gradient  force  per  meter;  that  is,  for  d.r  =  I. 

513.  r  =  JP  =  gj>mAB  in  terms  of  the  units  of  force  P. 

514.  /'  =  Ap  =  ffm JB  in  terms  of  the  units  of  weight  p. 
The  gradient  force  changes  with  the  temperature. 

Let  /'„  =  the  gradient  force  for  T0  =  273°  C.  and  Bn  =  0.760 

meter. 
/'  =  the  gradient  force  for  T  and  B. 

O1O.  ri IT    ni 

9   VT      W 
•*o    -° 

,        ,  laP  I  dp 

IV.  To  evaluate „     and : 

p  dx  f>  dx 

516.  We  have  P0  =  gj>mBn;  and  hence, 

517.  lf9P  1          dB  I     </« 

rt  n  Hot  n 


(1=  - 


0.0012G 


(G  is  in  meters.) 

518.  Also,  we  have  p0  =  <rmBn;  and  hence, 

519.  _  Idp_        I      dB  _        I       am 

~~pdx~     ~~p"mdx~     "^111  HI  G=  ~~p~ 

(G  is  in  meters.) 

IdP 
Numerous  other  evaluations  of -.—  are  given  in  Table  65, 

p.  489,  of  the  Cloud  Eeport. 

EVALUATION  OF    THE   GRADIENTS  IN  POLAR  COORDINATES. 

Since  the  angular  velocity  of  the  rotating  earth  is  n  sin  0  =  — , 

where  v'  is  the  absolute  eastward  velocity,  andr  =  6,370,191  +  h 

u'  cot  /y 

meters,  we  have  n  =  0. 00007292,  and  also  n  cos  0  =  ,  in 

T 

which  r  can  be  taken  practically  equal  to  R.     The  general  polar 
equations  of  motion  become,  by  substituting  these  values  in  181, 

194.  IdP      du      coift  „  uw 


IdP      dv 


IdP      dw 


r 
cotO 


w 


(2  v'  +  v)u  +  (2  «'+«)- 


---  ^—  =  -5T  —  (2  r'  +  v)  v  ---  h  g. 

P  dz       at          r  r 

The  terms  in  n2  which  give  the  figure  to  the  rotating  earth 
have  been  omitted,  and  the  inertia  terms  become  equal  to  zero 
for  steady  motions  of  the  atmosphere  ;  also,  for  all  except  compu- 
tations of  great  precision  the  terms  in  w  can  be  neglected. 

1  d  P 

To  evaluate  the  acceleration  —   —  ,  we  have,  first,  from  47, 

p  dx 


47«. 


_ 
P-P.P    Ta~  p  P(1 


IB     T  I 

=  t    jj    -,p  n  ,  for  variations  of  gravity,  since  g  =.'/„»,, 

'    O  O  1 

1  700  T     . 
=  —  _„    -  r    for  constant  gravity  and  B  in  nun. 

(t    alo  /> 


From  the  formulae  on  page  489, 
47/».   101'       I  'I  /:  <!  II       <! 

~p  dZmm79*f-d?*Dd'bu"dZ  '-  It 

1      .'/  ," 
=  -~  ^^j'  ^ 


G 
111 


*Vv  for  the  gradient  measured  in  meters 
J-Q  (?x  for  the  gradient  d\=  />'„—  A'in  nun 


520. 

521. 
522. 
523. 


13.5958 


760 


9.806 


~  0.00129305  x  273  x  11 1  111  1 1 1  x  II  > 

=  0.0025833  ~  Gs 

cot  (> 
= (2  i;'  -)-  f)  r,  by  equation  194.      Hence. 


B  cot  n 


and  similarly, 


,=  -  387.102  J  ~  (2  V  +  r)  ./, 
=  +  387.102  J  [  -~  (2  ,;'  +  ,-)  r  +  "'  -  r/]. 


Since  v'  is  a  function  of  <>,  that  is,  r' =  /*  rsintt,  these  terms 
can  be  computed  by  simple  tables,  such  as  those  in  Tables  104, 
105,  106,  of  the  International  Cloud  Report,  where  some  of 
the  terms  are  evaluated.  By  expressing  the  variation  of 

387.102  x  -jp  in  a  table  with  B  and   T  as  the  arguments,  the 

several  products  can  be  quickly  computed. 
Examples : 

I.  For  B  =  700  mm.  and  T=  260°  C., 


_ 
T 


2.6923; 


For  0  =  30°    north    polar    distance,   2»'  =  464.5   meters 
per  second; 

For  v  =  40  meters  per  second,  (2i/  +  c)  v  =  20,180. 


Hence,Gt5=  387.102  -^  co*  "  (-_V 


r)  r  =  5.71   millimeters 


per  111  111  meters. 

II.  This  latter  has  been  computed  from  the  tables  as  follows: 

378.102       =  1,042.2; 


'',  cot  0  .  2u  =  0.005052,  by  Table  104; 


cot   II 


.  v  .  v=  0.000435,  by  Table  106. 


The   sum  of    these   is  COt  °  (2v'  +  v)n  =  0.005487.      Hence 

n 

the  product,  Gx  =  1,042.2  x  0.005487  =  5.71  millimeters  per 
111  111  meters.  Similarly,  the  gradients  G f  and  G t  can  be 
computed. 

For  these  values  of  B  and  T,  we  find  in  other  examples, 

0  =  40°  0  =  50°      „        _  .O    0  =  60°  ,  «7 

= 


51 


We  are  at  last  in  a  position  to  examine  the  system  of  gradi- 
ents in  the  United  States  on  the  10,000-foot  plane  and  on  the 
3,500-foot  plane.  For  we  have  obtained  by  the  nephoscope 
aud  theodolite  observations  as  given  in  the  Cloud  Iteport  a 
large  number  of  corresponding  values  of  n  aud  r,  which  enter 
these  equations.  The  values  of  B  and  T  on  these  planes  have 
been  carefully  determined  for  each  month,  and  also  the  gra- 
dients by  which  such  values  can  be  determined  at  any  time. 
This  will  enable  us  to  discuss  the  effect  of  friction  at  these 
planes,  by  means  of  the  residuals  which  occur  between  the 
values  of  (f  as  found  by  these  formulfe  and  those  read  off 
from  the  charts  of  isobars  contained  in  the  Barometry  Report 
of  1900-1901. 

Furthermore,  our  Weather  Bureau  stations  will  soon  be 
provided  with  suitable  tables  for  computing  pressures  on  the 
3,500-foot  and  the  10,000-foot  planes,  and  this  will  give  daily 
configurations  of  isobars  on  these  two  levels.  If,  in  addition, 
we  had  measures  of  the  velocity  of  the  clouds,  q  (u,  v,  w),  above 
each  station  by  means  of  nephoscopic  observations,  it  would 
enable  vis  to  make  complete  dynamic  computations  of  the  forces 
acting  in  cyclones  and  anticyclones,  as  is  seen  by  an  inspection 
of  the  formulie. 

Since  the  tabular  computations  are  constructed  for  average 
conditions,  it  is  of  the  utmost  importance  that  check  ohxcn'a- 
li<ni*  be  made  on  those  two  planes  in  order  to  control  these 
(Ivnamic  discussions  and  make  them  more  perfect.  Such  ob- 
servations can  be  made  by  balloon  ascensions  up  to  2  or  3 
miles,  or  by  kite  ascensions  up  to  10,000  feet,  or  by  certain 
computations  on  cloud  observations. 

It  seems  to  me  very  clear  that  a  series  of  suitable  research 
explorations  would  soon  result  in  placing  our  dynamic  meteor- 
ology upon  a  satisfactory  scientific  basis,  and  put  an  end  to 
the  fruitless  speculations  which  have  done  so  little  to  advance 
our  knowledge  of  the  laws  of  the  atmospheric  motions. 

THK    EQUATION    OF    CONTINUITY,   AND    SOME    DERIVED    RELATIONS. 

1.  The  equation  of  continuity  can  be  found  as  follows: 
Consider  a  cylinder  of  the  height  z  and  radius  m,  into  which 

air  of  the   density  //  streams  equally  from  all  sides  with  the 

velocity  —  it,  since  the  direction  is  negative. 


-u 


FIG.  22.— Illustrating  the  formation  of  the  equation  of  continuity. 
The  amount  of  instreaming  air  in  the  unit  of  time  is 

—  2-iu  z  iif>. 

If  the  air  is  incompressible,  then  there  will  stream  into 
a  cylinder,  whose  radius  is  smaller  by  dm,  the  amount 
—  2*  (m  —  dm)  zu/i,  and  at  the  same  time  there  will  escape 
upward  between  these  two  cylinders  the  amount  2n-nj  dm 
Hence 

524.  —  1-m  z  Uf>  +  2-  (m  —  dm)  z  v./i  =  —  1-dm  sup 

=  Ixmdm  w/>. 

Integrating  along  the  entire  radius  from  0  to  m,  we  have, 

r™  ra 

—  2-/<  I      zudw  =  2-/»   I       mdm  w. 

Jo  Jo 


Therefore,  —  zum  —  t  m'*w,  and  the   equation  of  continuity 
2 


becomes 

487. 


—  2«3  =  mw. 


This  applies  to  pure  vortex  motion,  and  it  finds  some  ex- 
amples in  the  atmosphere,  such  as  in  tornadoes,  in  many  hurri- 
;anes,  and  in  some  highly  developed  cyclones. 

It  may  be  remarked  that  in  treating  of  the  general  circula- 
tion of  the  atmosphere,  the  application  of  the  pure  vortex  law 
=  constant,  has  failed  to  give  correct  results,  for  example 
in  the  writings  of  Ferrel,  von  Helmholtz,  Oberbeck,  Sprung, 
and  others.  This  leads  to  the  theory  of  contracting  rings  on 
the  earth  with  progressive  motion  towards  the  poles,  or  ex- 
panding rings  with  progression  towards  the  equator.  While 
the  law  of  the  sum  of  the  momenta  -  mv  =  0  must  prevail,  the 
rings  are  in  nature  broken  up  into  such  complex  stream  lines 
as  to  render  integration  by  the  simple  vortex  law  too  rough 
and  ready  a  method.  We  must,  therefore,  study  the  theory 
of  typical  stream  lines,  before  attempting  any  general  inte- 
gration for  the  entire  circulation. 

The  following  derived  relations  are  convenient: 


2.  Since  —  2«2=  raw,  we  have 


w 

=  __  GJ 
2z 


=   '    m,  if  ID  =  cz. 


3.  For  sand  —  u  both  constant,  we  have  rszu  =  —  const.  =  <,'>. 

Hence,  *  =  -  C°nst-  =  -  f  C°™i-\   -\,  and  by  differentiation, 
uts  \    <;/2    J  nr 

const. 


dz=  + 


r/2 


=  _22  — .     Therefore, 

CT 


dz  dta      ,  dz  n  drs 

-  =  —  2        ;  also,  ra       —  _  23-,.  . 
ra  dt  at 


4.  These  give  the  form  for  the  current  function  <j>,  and  the 
velocity  potential  <f,  in  two  cases. 


488. 


489. 


I.  </'  =  »  ras  =  —  -  -  ra2  z  = 


a  = 


k — c 


II.  4'  =  —  cz. 


<?=  —c. 

I 

tt=k- 


5.  If  the  current  function  is  modified  through  a  deflecting 
force  and  also  through  friction,  then  the  equation  of  motion 
has  two  solutions,  so  that  the  roots  of 


dv       uv 


are,  by  438, 


A       c 


Ac 


»m=-w~cu. 


In  obtaining  the  velocities  of  the  rotation,  c,  we  can  modify 
the  current  function,  as  follows,  namely,  multiply  by 


a  = 


-,  —     . 

K  —  C 


. 
f  =      in  the  two  cases. 

' 


52 


6.  Hence,  by  using  Stokes  's  current  functions,  we  find  for 
the  velocities  n,  r,  w  in  the  two  cases, 


490. 


Case  I. 


JL^  Of       <p  _         c 
r-~      =      =     ~ 


a</'  /.s       /• 

v  =  4-  —  =  —  ,  ro  ==  4- 

ra  /•  —  c   2  T 


2o  k  —  c, 
m       /. 


Case  II. 


1  04'     r  c 

M   =   4-  -     =         =   — 

ta  oz       w  m 


v  =  + 


/I  c 


7.  In  unconstrained  motion  the  vortex  law  of  preservation 
of  areas  is 

c,  tnz  ZZJ'K; 

i>ro  =s  —  o  m2z=  2r/        =  muz=  —    o     =  constant, 

m    '  V  u 

by  307,  and  by  308,  introducing  the  value  V*  =  %gz. 

This  vortex  law  when  modified  by  deflection  and  friction 
becomes, 

'  492.         Case  I. 


Case  II. 


i       c 

k  —  c  2 

A 


*z  =  const. 


cz  =  const. 


8.  The  inclination  of  the  stream  line  to  the  isobars  is, 
491.          Case  I. 


cot  i  =  + 


k-c' 


Case  II.  /i 

C0ll=    +  T*. 

k 

9.  The  equation  of  continuity  (163)  is  satisfied  by  the  values 
in  490. 

493.          Case  I.     du        u       div 

f)w        w        Oz 

Case  II.    du        u       dw 


c       c 
m  +  Oz  =  ~  2  ~~  2  +  c  = 


10.  The  equation  for  gradient  has  a  term  to  express  the  un- 
evaluated  variation  due  to  temperature  effects,  f(tx),  and  it 
becomes,  for  the  radial  component, 

494.  _!>•  „  1  OP 

,,  ^'*  ~        /,  Ox 


=  —  2  ra  f  &•—  c  +  /  c'ot  i  +  g  cot2 
11.  The  total  velocity  is 


i  1 


. 

sm2  i  '  sm 

12.  The  variation  of  pressure  can  be  expressed  by 


These  formulsc  are  all  collected  in  Table  121,  page  G02,  of  my 
International  Cloud  Report. 

This  system  of  formula  applies  directly  only  to  the  pure 
vortex  motions  that  satisfy  the  assumed  current  function  and 
velocity  potential.  The  components  u,  v,  iv,  are  so  simply 


interrelated  that  it  is  usually  possible  to  make  enough  ob- 
servations of  some  sort  from  which  to  derive  all  the  other 
vortex  relations.  Applications  of  them  were  made  in  the  In- 
ternational Cloud  lleport  to  two  cases;  (1)  The  waterspout 
observed  off  Cottage  City,  Marthas  Vineyard,  Mass.,  August 
19,  1896,  on  page  (i.'t:i;  upon  this  important  formation,  a  fuller 
report  will  be  published.  (2)  The  average  velocities  in  a  cy- 
clone from  the  data  in  Table  12(5,  as  given  on  page  629  of  the 
International  Cloud  Report.  The  outcome  of  these  computa- 
tion*! is  to  show  that  the  natural  stream  lines  of  the  atmosphere 
conform  on  the  average  to  these  formulae.  There  are,  however, 
wide  divergences  of  such  a  type  as  to  indicate  that  the  pure 
vortex  motion  is  seriously  modified  by  several  conflicting  forces, 
and  that  the  true  problems  for  the  meteorologist  consist  in 
discovering  the  nature  and  amount  of  these  deviations  of  tlio 
currents  of  the  atmosphere  from  the  simple  laws.  This  is  in 
fact  a  task  of  great  difficulty,  but  it  has  now  become  evident 
what  should  be  the  course  of  scientific  development  for  meteor- 
ology. There  is  little  use  in  a  further  discussion  of  the  gen- 
eral theorems  at  the  present  time,  but  there  is  great  need  of 
procuring  the  right  kind  of  observations  for  use  in  such  prob- 
lems. The  Weather  Bureau  has  accordingly  been  engaged  in 
such  a  reconstruction  of  its  data  as  will  contribute  to  the  solu- 
tion of  these  problems  for  the  United  States.  We  have  already 
published  a  large  number  of  nephoscope  velocities  for  the 
eastern  half  of  the  country;  the  velocities  of  the  upper  cur- 
rents for  the  West  Indies  have  been  determined  for  about  three 
years,  July  1899-July  1902,  and  their  computation  will  be  com- 
menced at  once;  similar  nephoscope  observations  will  be  un- 
dertaken for  the  Rocky  Mountain  and  Pacific  districts,  begin- 
ning about  July  1902.  Our  barometric  observations  have  been 
thoroughly  reduced  for  the  years  1873  to  the  present  time,  and 
the  tables  necessary  for  reductions  to  the  three  reference  planes 
are  in  hand  for  the  construction  of  daily  maps  at  three  levels, 
containing  the  system  of  isobars  corresponding  with  them.  It 
will  be  necessary  to  revise  the  temperature  and  vapor  tension 
observations  and  reduce  them  to  homogeneous  systems  before 
our  data  will  be  complete  for  the  application  of  the  theoretical 
equations  to  the  observational  data.  It  is  desirable  to  put  an 
end  to  general  mathematical  speculation  in  meteorology,  and 
to  substitute  for  it  definite  comparisons  between  observations 
and  computations  together  with  dependent  solutions  for  the 
outstanding  unknown  quantities. 

THE  PROBLEMS  OP  THE  AQUEOUS  VAPOR  CONTENTS  OF  THE  ATMOSPHERE. 

I  shall  allow  myself  only  a  few  remarks  regarding  the 
methods  which  were  used  in  my  report  for  the  discussion  of 
the  various  complicated  problems  that  concern  the  aqueous 
vapor  contents  of  the  atmosphere,  because  the  details  are  too 
complex  for  a  brief  summary  like  this,  and  also  because  the 
work  was  given  in  such  an  extended  form  as  to  enable  students 
to  follow  it  without  difficulty.  There  are,  however,  a  few 
leading  ideas  to  which  attention  may  be  especially  directed, 
as  they  serve  for  an  introduction  to  the  subject  in  general. 

There  is  collected  in  the  International  Cloud  Report,  Table 
64,  "Fundamental  constants,"  a  series  of  elementary  constants 
in  the  English  and  metric  systems,  with  the  logarithms  of  the 
constants,  and  also  a  set  of  elementary  formulfe  which  are 
most  useful  in  meteorological  studies.  They  cover  nearly  all 
the  simple  relations  which  constantly  recur  in  manifold  forms 
in  the  treatises  and  papers  on  meteorological  subjects,  and  by 
transformation  and  combination  a  multitude  of  different  rela- 
tions can  be  readily  obtained.  Tables  63,  64,  and  65  supply 
the  basis  for  much  descriptive  matter  commonly  found  in 
treatises,  in  so  compact  and  accurate  a  form  as  to  quite  super- 
sede the  lengthy  statements  with  which  the  same  laws  are 
usually  presented,  and  this  is  a  great  convenience  for  the 
student  and  computer.  Those  who  will  take  the  trouble  to 
become  familiar  with  these  tables  will  find  much  saving  of 


53 


time  in  general  work,  and  also  they  will  be  guarded  from 
such  errors  of  thought  and  statement  as  are  likely  to  occur 
from  not  having  these  formula:  in  mind,  or  accessible  for  con- 
venient reference. 

Iii  treating  the  vapor  problems  I   have  referred  all  the  for- 

g 

mulao  to  the  ratio  -w,  vapor  tension  divided  by  barometric  pres- 
sure, as  the  most  convenient  and  accurate  argument  for  com- 
bination with  another  argument,  as  the  height  h,  the  tempera- 
ture T,  or  the  pressure  B.  The  Table  07  summarizes  the  for- 
mulic  for  the  hypsometric  reductions,  and  they  are  more  fully 
explained  in  the  forthcoming  Barometry  Report.  The  general 

en 
idea  is  that  having  found  the  ratio  ,"  at  the  base  of  a  column, 

**0 

the  application  of  Hann's  law  for  the  diminution  of  the  vapor 
pressure  with  the  height  gives  the  most  accurate  average  law 
for  computing  the  integral  of  the  vapor  tension  throughout 
the  entire  column.  A  small  secondary  term  can  be  added 
whenever  our  knowledge  of  the  facts  justifies  such  an  increased 
degree  of  accuracy,  though  it  is  usually  of  little  importance, 
especially  for  a  series  of  observations  where  mean  results  are 
required. 

In  the   development  of  the  a,  /?,  f,  ft  stages  of  the  adiabatic 

thermodynamic  formulae,  the  ratio  jr  is  made  the  primary  ar- 
gument by  the  series  of  transformations  given  in  Table  72. 
These  formulae  are  reduced  to  numerical  tables,  94-102,  and 
their  accuracy  is  tested  by  comparing  directly  with  the  Hertzian 
logarithmic  formulae,  as  given  in  the  examples  of  Table  108. 
Their  use  involves  a  series  of  solutions  by  trials,  which  though 
laborious,  yet  lead  to  perfectly  rigorous  results,  and  after  a  little 
practise  it  becomes  quite  easy  to  obtain  the  true  trial  values 
without  much  difficulty.  The  graphical  diagrams  of  Hertz 
give  only  approximate  values,  because  they  throw  out  the 
vapor  tension  term  in  the  critical  places  and  thus  render  in- 
accurate the  very  problems  they  were  designed  to  discuss. 
Special  applications  were  made  to  finding  the  gradients  of 
pressure,  temperature,  and  vapor  tension  in  the  «,  ft,  f,  K 
stages,  and  the  results  are  found  in  Tables  147  for  metric 
measures,  and  in  Tables  153  for  English  measures. 

Finally  the  same  tables  were  employed  to  discuss  the  impor- 
tant problem  of  the  difference-  between  an  adiabatic  atmos- 
phere and  the  one  given  by  the  upper  strata  observations, 
whereby  a  new  method  was  illustrated,  with  results  in  Table 
162.  The  value  of  this  computation  depends,  of  course,  upon 
the  data  1>,  T,  e,  adopted  for  the  upper  atmosphere,  as  meas- 
ured by  the  balloon  and  kite  ascensions.  It  was  especially 
necessary  to  have  the  temperatures  at  high  levels,  and  for  this 
purpose  I  collected  such  material  as  was  available  up  to  the 
end  of  the  year  189G,  when  I  began  this  compilation,  and  for 
that  purpose  employed  the  102  balloon  ascensions  enumerated 
in  Table  155,  embracing  all  those  then  available  for  the  United 
States,  England,  France,  Germany,  and  Russia.  I  expressed 
myself  cautiously  regarding  the  result,  page  750,  holding  the 
computation  as  preliminary  to  a  fuller  one  which  would  be- 
come possible  when  accurate  observations  had  been  accumu- 
lated for  the  upper  air  temperatures,  and  I  have  therefore  had 
an  interest  in  examining  the  Berlin  report  of  the  German  bal- 
loon ascensions.*  In  the  first  volume  of  this  work  are  contained 
the  data  for  each  ascension,  and  in  the  Meteorologische  Zeit- 
schrift,  October,  1901,  page  449,  H.  Hergesell  gives  a  summary 
of  the  resulting  free  air  temperatures.  I  have  extracted  the 
observed  temperatures  from  this  report,  interpolated  them  to 
each  round  1,000-meter  level,  and  computed  the  total  tempera- 
ture fall  from  the  surface  to  the  respective  strata,  with  the 
result  given  in  Table  21  and  fig.  23.  If  the  ascensions  are 
divided  into  three  sets,  A,  those  reaching  heights  between  the 


A. 

B. 

c 

D. 

E. 

F. 

G. 

II. 

/. 

18080 

—  60  4 

—68  3 

—71.1 

—115  o* 

15000 

59  1 

6G  0 

68  0 

1011  0* 

14000 

—  00  9 

—62  5 

—04.  7 

—  97.0* 

13000 

60  1 

63  5 

—  61.0 

—  88.  0» 

12000 

61  0 

60.3 

—57.0 

—  79.0* 

1  1000 

—62  8 

52  7 

—  52.6 

—  70.0* 

10000 

60  6 

18  5 

—  60 

—62 

—48.1 

—  61.  Of 

9000 

•ix  d 

—57.0 

—44.6 

—56.8 

—51 

—  56 

—43.  4 

—  54.  5-( 

8000 

47  4 

51  0 

34  9 

48  7 

—47 

—48 

—38.  5 

—  47.  9} 

7000 

—38.4 

—  44.8 

—31.7 

—39.8 

—38 

—41 

-33.8 

—  39.  6-f 

6000 

32  0 

37  5 

26  9 

34  6 

—30 

—  34 

—28.1 

—  32.9 

6000 

20  8 

—25  5 

—32.3 

—23  1 

—27.0 

—25 

—26 

—22.8 

—  26.04 

4000 

—15.0 

—19.  6 

—28.  0 

—19.0 

—20.7 

—18 

—21 

—17.9 

—  19.9 

3000 

12.9 

14  3 

19  5 

—13.0 

—  15.4 

—13 

—16 

—13.1 

—  14.51 

2000  .             

—  7.9 

—  8.5 

—15.8 

—  9.6 

—  9.9 

—  9 

—  8 

—  7.8 

—    9.0- 

1000 

3  2 

3.7 

8.3 

—  3.8 

—  5.0 

—  4 

_  4 

—  3.9 

—    4.3- 

0000  . 

0.0 

0.0 

0.0 

0.0 

0.0 

0 

0 

0.0 

0.0 

'  Wissi'iisrlial'llirhr   Luftfahrten. 
Berlin,  18'Ji). 


AsKiiinim    iinil    I>ITSOH.     3 


surface  and  5,000  meters,   7?,  those  between  the  surface  and 
10,000  meters,  and   C,  those  between  the  surface  and  10,000 

TABLE  21. — Comparison  of  several  determinations  of  the  total  temperature 
change  from  the  surface  to  high  levels. 


A  =49  ascensions  not  above  5,000  meters  in  manned  balloons. 

./?  =  12  trips  upward  and  5  downward,  not  above  10,000  meters,  in  manned  balloons. 
C"=9  ascensions  of  unmanned  balloons  above  10,000  meters. 
/>=  Bigelow's  compiled  data.  Tables  156,  I.  IT.,  International  Cloud  Report.. 
E=  Bcrson's  mean  results,  Meteorologische  Zeitschrift,  Oct.  1901,  p.  449. 
,F=Teisserenc  de  Bort's  mean  results,  Meteorologische  Zeitschrift,  Oct.  1901,  p.  449. 
G  =  HergeseH's  mean  results.  Meteorologische  Zeitschrift.  Oct.  1901   p.  449. 
Jf=  Higelow's  mean  results,  Tables  157,  I.  II.,  International  Cloud  Report. 
7=The  mean  of  E,  F,  G  up  to  10,000  meters,  and  a  gradient  of  9°  per  1,000  meters  from 
11,000  to  16,000  meters. 

*  Hergesell's  assumed  gradient  9°  per  1, 000  meters, 
t  Mean  of  E,  F,  G. 


ffft 

mfftem 

j                    jn>  c 

\ 

\ 

\ 

•\ 

14000 
73000 
72OOO 

i7ooo 
roooo 

90OO 
8OOO 
7OOO 
6000 
6000 
400O 
3OOO 
2000 

7C0O 
O 

\ 

\  I 

i 

\ 

t 

\ 

• 

N 

lh 

V 

\ 

v 

\ 

( 

\ 

V 

^ 

\ 

\. 

\\ 

\ 

\ 

^ 

\ 

\ 

\ 
\ 

\ 

\ 

^ 

\ 

\ 

B 

\ 

^ 

\ 

$ 

\1 

4 

\ 

\ 

^ 

\ 

\ 

\ 

^ 

\ 

\ 

^ 
\ 

\ 

\ 

\ 

Temp    H0°    700°    ffO°     60°    70°     ffO'    SO"    4O°    30'    2O°    7O'     0 
faii,C° 

A,  and  B.  Berlin  observations  with  manned  balloons. 
C.  fli-rliiL  oliM-rvations  with  unmanni'il  balloons. 
/>.  Ili^elow's  summary  from  all  countries. 
//.   Bigelow'a  adopted  mean  result.. 
7.  Itcrlin  adopted  mean  result. 

KK;.   'J3. — Total   temperature  fall  from  the   surface  to  high   levels 
several  systems, 


54 


meters,  we  have  the  following  remarkable  data.  Class  A  con- 
tains 49  ascensions  of  manned  balloons,  and  gives  a  tempera- 
ture fall  of  20.8°  at  the  5,000-meter  level;  class  11  contains  12 
upward  and  5  downward  trips  of  manned  balloons  and  gives  a 
fall  of  25.5°  at  the  5,000-meter  level,. or  5°  more  than  cluss  .  I ; 
class  G  contains  12  ascensions  of  unmanned  balloons,  with  a 
fall  of  32.3°  at  5,000  meters,  or  11.5°  more  than  in  class  A,  and 
57°  at  9,000 meters,  or  9°  more  than  in  class  B.  This  class  shows 
also  a  fall  of  60.6°  at  10,000  meters  and  G0.4°  at  16,000  meters. 
These  widely  different  temperature  falls  by  classes  A,  B,  G 
may  possibly  be  explained  by  those  who  are  familiar  with  the 
circumstances,  but  the  fact  deserves  attention;  also  the  other 
fact  that  there  is  no  temperature  fall  between  10,000  and 
16,000  meters  as  observed  in  the  Berlin  unmanned  balloon 
ascensions.  In  column  D  is  given  the  result  of  my  own  com- 
pilation found  by  taking  the  mean  of  all  the  figures  as  they 
stand  in  Tables  156,  I,  II;  and  on  fig.  21  the  line  D  is  seen  to 
fall  between  A  and  H  and  to  cross  G  at  the  height  of  12,000 
meters. 

In  his  review  of  the  Berlin  ascensions  H.  Hergesell  gave  the 
Berson  results  as  shown  as  in  column  E,  the  Teisserenc  de  Bort 
results  as  in  column  F,  and  his  own  results  as  in  column  G.  He 
also  stated  the  conclusion  that  above  10,000  meters  the  adia- 
batic  rate  of  temperature  fall  in  free  air  prevails,  and  this  may 
be  considered  as  9.0°  per  1,000  meters,  as  suggested  by  him. 
Column  /  is  the  mean  value  of  E,  F,  G,  up  to  10,000  meters, 
and  from  that  level  to  16,000  the  fall  is  calculated  at  9.0°  per  | 
1,000  meters,  these  values  being  plotted  on  fig.  21.  Finally, 
by  taking  the  means  of  the  data  given  in  Tables  157,  I,  II, 
which  was  derived  from  Charts  78,  79,  as  constructed  to  de- 
termine the  gradients  for  each  month  in  the  year,  we  have  the 
data  of  column  H,  also  plotted  on  fig.  21.  It  is  seen  that  my 
adopted  result,  H,  lies  midway  between  A  and  B,  and  is  a  fair 
average  of  all  the  ascensions  taken  in  the  unmanned  balloons, 
while  the  adopted  Berlin  result,  /,  is  45°  lower  at  16,000 
meters,  giving  at  that  level  a  temperature  of  — 115°  approxi- 
mately. There  is  a  further  consideration  of  importance  to  be 
noted  in  this  connection.  E.  Rogovsky  in  his  paper  on  the 
"Temperature  and  composition  of  the  atmospheres  of  planets 
and  the  sun,"  Astrophysics,  November,  1901,  discusses  the 
temperature  of  the  interplanetary  medium  (according  to 
Pouillet  —142°  C.,  Froelich  —131°  to  —127°),  and  assumes 


it  to  be  — 142°  C.  A  fair  supposition  regarding  the  efficient 
depth  of  the  atmosphere  makes  it  04,000  meters  or  about  40 
miles,  and  hence  we  have  the  following  data: 


llfiulii  cii 

;it  iih'.-pln  1  1 

Blgelow. 

IVrlin. 

Ti'iiiprratiirr. 

Neoenarj 

HKi'lirlll-. 

Tempermtara. 

Nrrr>s:iry 

gradient*. 

MMorv, 
14,000 

16,000 
Sitrl'iice    

°  C. 
—142 

—  55 
15 

0  a 

0  a 
ua 

C 

—1.8 

—ii.  11 

—100 

—4.4 

--7.2 

15 

If  the  temperature  falls  from  15°  at  the  surface  to  —  ~>~> 
at  16,000  meters  with  a  gradient  of  about — 4.4°  per  1,000 
meters,  then  to  reach  — 142°  at  64,000  meters  the  gradient 
should  on  the  average  be — 1.8°.  It  will  be  seen  }>\  mv  Charts 
78  and  79,  International  Cloud  Report,  that  I  adopted  an  in- 
creasingly slower  temperature  fall  with  the  height  in  the 
strata  above  10,000  meters,  in  accordance  with  this  general  view. 
If  the  Berlin  theory  is  assumed  that  a  fall  of  9.0°  per  1,000 
meters  prevails  above  the  10,000-foot  level,  then  it  must  some- 
where rapidly  decrease  to  a  very  small  gradient  in  order  not  to 
diminish  the  exterpolated  temperatures  far  below  that  value 
assigned  by  certain  astrophysicists  to  the  celestial  medium  at 
the  earth's  distance  from  the  sun.  In  fact  the  gradient  becomes 
one-tenth  of  the  adiabatic  rate,  which  was  actually  assumed. 
If  the  temperature  • — 260°  C.  is  that  of  the  interplanetary 
medium,  as  supposed  by  other  writers,  these  inferences  must 
be  modified  accordingly. 

From  these  two  cousiderations,(l)  that  my  temperature  syst  em 
includes  the  data  of  the  highest  balloon  ascensions,  and  (2) 
that  my  gradients  are  in  harmony  with  the  requirements  of 
astrophysics,'  I  shall  let  my  computations  on  the  heat  differ- 
ence between  the  adiabatic  and  the  actual  atmosphere  stand 
as  they  were  given  in  my  report.  The  accurate  measurement 
of  the  temperatures  in  the  highest  strata  is  a  very  difficult 
process,  and  all  efforts  to  secure  reliable  results  deserve  the 
hearty  support  of  meteorological  physicists.  There  are  several 
problems  whose  solution  depends  upon  the  possession  of  such 
data  in  a  satisfactory  form. 


VII.— A  CONTRIBUTION  TO  COSMICAL  METEOROLOGY.1 


GENERAL    REMARKS. 

I  have  already  published  the  results  of  certain  computations 
and  discussions  on  the  subject  of  the  direct  connections  between 
the  variations  of  the  solar  output  of  energy,  and  the  correspond- 
ing synchronisms  in  the  meteorological  elements  of  the  earth's 
atmosphere.  These  are  in  particular,  Solar  and  Terrestrial 
Magnetism,  Weather  Bureau  Bulletin  No.  21, 1898,  and  Eclipse 
Meteorology  and  Allied  Problems,  Weather  Bureau  Bulletin  I, 
1902,  which  include  the  substance  of  other  minor  papers 
related  to  this  subject.  The  purpose  of  these  studies  has  been, 
(1)  to  establish  the  fact  that  a  synchronous  connection  does 
exist  between  the  solar  and  the  terrestrial  forces,  and  (2)  to 
derive  the  operation  of  these  periodic  movements  so  as  to  ulti- 
mately lead  meteorology  to  a  scientific  understanding  of  the 
terrestrial  seasonal  climatic  changes,  and  to  a  true  basis  for 
forecasts  of  weather  conditions,  at  least  one  year  in  advance. 

The  difficulty  of  reaching  a  correct  solution  of  this  problem 
is  well  understood  by  those  who  have  worked  upon  it,  to  reside 
in  the  unsteadiness  of  the  solar  output  itself,  and  the  numer- 
ous subordinate  transformations  of  the  energy,  through  the 
radiation,  the  general  and  local  cyclonic  circulations,  till  it 
culminates  in  a  season  having  certain  characteristics.  The 
material  for  the  study  consists  in  the  variations  of  the  pressures, 
temperatures,  and  vapor  tensions  at  many  stations  in  different 
portions  of  the  earth,  in  the  fluctuations  of  the  terrestrial  mag- 
netic field,  in  the  changes  of  the  spectrum  energy  of  the  solar 
and  the  aqueous  vapor  curves,  and  in  the  variations  of  the  sun 
spots,  the  prominences,  and  the  solar  faculre.  The  magnitude 
of  the  task  involved  in  handling  this  material  is  such  as  to 
limit  the  attempt  to  deal  with  it  to  a  few  institutions  having 
these  subjects  specially  in  charge.  Among  them  the  United 
States  Weather  Bureau  has  been  able  to  make  some  contribu- 
tions from  time  to  time. 

SUMMARY    OF    THE    DISCUSSION    OF    1898. 

On  pages  121-130,  Bulletin  No.  21,  is  given  a  brief  account 
of  an  extensive  discussion  of  the  data  then  at  hand,  and  the 
result  was  such  as  to  show  that  there  is  a  marked  synchronism  i 
between  the  solar  and  terrestrial  variations  of  energy.  Fig.  | 
24  serves  to  recall  this  fact  and  it  shows  that  in  the  sun-spot 
period,  1878-1893,  there  was  a  true  synchronism  in  the  varia- 
tion of  the  sun-spot  areas,  the  European  magnetic  force,  which 
is  the  resultant  of  the  two  components  measured  on  a  horizon- 
tal plane,  and  the  American  meteorological  system.  The  latter 
includes  a  variation  of  temperature  at  25  stations  in  the  north- 
western portions  of  the  United  States,  the  pressure  at  10 
stations,  the  variable  mean  movements  of  the  storms  in  latitude 
and  longitude,  and  the  movement  of  the  tracks  of  the  cold  | 
waves  in  latitude.  Each  of  the  two  latter  elements  was  derived 
from  an  exhaustive  compilation  of  the  coordinate  positions  of 
the  cyclonic  centers  for  the  interval  of  fifteen  years  ending 
with  1893.  It  led  me  to  the  following  summary: 

The  increase  of  solar  magnetic  intensity  is  synchronous  with  a  diminu- 
tion of  temperature  but  with  an  increase  of  pressure,  and  this  function  i 
persists  throughout  every  phase  of  the  research. 

In  spite  of  some  irregularity,  there  is  a  distinct  conformity  in  the  gen- 
eral sweep  of  these  curves,  and  also  in  the  tendency  to  describe  crests 
d  iii-ing  the  same  years.  Indeed,  the  occurrence  of  four  subordinate  crests 
in  the  11-year  period  suggests  strongly  that  a  2 j-year  period  is  superposed 
upon  the  long  sweep  of  that  periodic  curve.  Apparently  this  minor 
period  is  the  basis  of  these  seasonal  variations  of  the  weather  condi- 
tions of  the  United  States  more  than  anything  else,  so  that  in  long-range 
forecasting  this  period  must  be  very  carefully  considered. 

It  was  for  the  purpose  of   carrying   this  subject  one  step 
1  Reprinted  from  the  Monthly  Weather  Keview  for  July,  1902. 


further  forward  that  the  discussion  of  the  data  summarized  in 
this  present  paper  was  undertaken.  There  has  been  consider- 
able delay  in  completing  the  work  on  account  of  many  other 
important  duties. 

It  is  evident  that  the  terrestrial  magnetic  field  affords  the 
data  which  is  most  available  for  studying  the  fundamental 
periods  in  this  solar-terrestrial  synchronism.  An  exact  quan- 
titative computation  for  the  several  elements  involves  a  very 
large  amount  of  labor,  and  therefore  it  is  important  as  an 
alternative  to  derive  the  periods  by  methods  which  shall  give  re- 
liable proportional  variations  of  the  elements.  The  ideal  treat- 
ment is  to  compute  the  total  deflecting  force  of  the  magnetic 
field,  by  using  the  means  of  24-hourly  observations  of  the 
horizontal  force,  the  declination,  and  the  vertical  force,  taking 
out  their  daily  component  variations  in  rectangular  coordinates 
(dx,  dy,  dz,)  and  combining  them  into  polar  coordinates  s,  «,  /?. 


of 
/Sun   Spots. 


3,000 
2000 
JflOO 


MeanAateric 


Elements  o 


90 
SO 
70 


+JO 

o 

-JO 


mencan 


TraxA 
movement   in 


Storm  TracA- 
movement  in 


47° 

48 


.00 


4.OO 
4.20 
4.4O 


a.  so 

3.00 
S.SO 


3*90 
S.9O 
3.70 


feteorological  System 


ZN 


FIG.  24. 

The  next  simpler  method  is  to  omit  the  vertical  component, 
as  one  is  tempted  to  do  in  consequence  of  the  unreliable  action 
of  the  Lloyd's  balance,  and  turn  the  horizontal  components, 
dx,  dy,  into  polar  coordinates  IT,  ft,  on  the  horizontal  plane. 


56 


Since  it  has  been  proved  l>y  computation  that  the  east-west 
component,  dy,  derived  from  variations  of  the  declination, 
practically  disappears,  as  it  should  do  by  theory,  we  may 
adopt  the  variations  of  the  horizontal  component,  dx,  along 
the  magnetic  meridians,  as  the  best  single  component  for  com- 
putation. This  is  all  the  more  satisfactory  because  the  bililar 
horizontal  magnet  is  the  most  efficient  instrument  in  use  in 
the  magnetic  observatories,  and  is  generally  free  from  ob- 
jectionable features  in  its  operations.  I  have,  therefore, 
in  this  discussion,  adopted  the  variations  of  the  horizontal 
force,  as  shown  by  the  24-hourly  means,  on  the  ground  that 
they  are  proportional  to  the  total  variations  of  the  magnetic 
field  and  quite  free  from  instrumental  errors. 

THE    MAGNETIC    OBSERVATIONS   1841-1899. 

Accordingly,  the  magnetic  horizontal  force  for  the  interval 
1841-1899  has  been  submitted  to  a  discussion,  the  result  of 
which  is  summarized  in  this  section.  The  synchronous  action 
of  the  solar  energy,  as  exhibited  in  the  variation  of  the  sun 
spots,  the  terrestrial  aurora,  the  magnetic  field,  and  several 
other  phenomena,  has  been  frequently  developed,  so  that  the 
general  fact  is  admitted  by  all  students,  but  it  is  now  im- 
portant to  trace  out  this  sympathetic  movement  in  these  cos- 
mical  forces  more  in  detail,  especially  as  they  relate  to  the 
annual  and  seasonal  variations  in  the  earth's  atmosphere.  In 
the  Proceedings  of  the  Koyal  Society,  volume  63,  Mr.  William 
Ellis,  F.  R.  S.,  has  exhibited  this  synchronism  between  Wolf's 
sun-spot  numbers  and  the  declination  and  horizontal  force  at 
the  Greenwich  Observatory  for  the  interval  1841-1896.  (Com- 
pare Bulletin  I,  1902,  page  105.)  In  his  diagram  not  only  do 
the  curves  present  the  same  large  sweeps,  but  also  the  minor 
variations  appear  simultaneously  in  the  three  curves.  It  is 
for  the  purpose  of  developing  yet  more  distinctly  these  minor 
fluctuations  that  the  compilation  of  the  following  magnetic 
observations  was  executed. 

Instead  of  confining  the  study  to  a  single  observatory,  it  lias 
been  extended  so  as  to  practically  include  the  entire  earth,  at 
least  sufficiently  to  demonstrate  that  the  variations  are  com- 
mon to  the  whole  terrestrial  magnetic  field.  Thus,  for  dif- 
ferent years  we  studied  the  records  at  the  following  stations: 

1841-44.   Toronto,  St.  Helena,  Hobarton." 

1845.  Greenwich,  Toronto,  Singapore,  St.  Helena,  Cape  of 
Good  Hope,  Hobarton. 

1846^7.  Toronto,  St.  Helena,  Hobarton. 

1848.   Toronto,  Greenwich,  Hobarton. 

1849-1870.  Whatever  was  available,  as  Greenwich,  Toronto, 
Madras,  Batavia,  Pavlosk,  some  of  the  data  being  unsatisfactory. 

1871-77.  Greenwich,  Pavlosk. 

1878-1885.  Greenwich,  Pavlosk,  Vienna,  Prague,  Tiflis. 

1886-1887.  Los  Angeles,  Toronto,  Greenwich,  Paris,  Pola, 
Prague,  Pavlosk,  Tiflis,  Zi-ka-wei,  Batavia. 

1888.  Greenwich,  Prague,  Pavlosk. 

1889-90.  Greenwich,  Washington,  Pavlosk. 

1891.  Greenwich,  Prague,  Pavlosk. 

1892-99.  Paris,  Pola,  Pavlosk. 

By  thus  changing  the  stations  it  becomes  impossible  that 
the  peculiar  action  of  any  set  of  instruments,  should  such  ex- 
ist, can  impose  a  bias  upon  the  final  result.  It  seems  to  me 
that  it  makes  no  difference  what  three  stations  are  chosen  to 
represent  the  cosmical  variation  of  the  magnetic  field,  as  indi- 
cated by  the  horizontal  force  which  is  proportional  to  the  total 
force.  Three  stations  are  desirable  in  order  to  eliminate  the 
local  impulses  of  the  field,  and  if  they  had  been  available  I 
should  have  used  the  same  three  stations  throughout,  all  re- 
duced to  the  C.  G.  S.  system  of  units,  for  the  sake  of  having 
rigorous  quantitative  results.  The  data  of  this  paper  limit  it 
to  showing  relative  synchronisms,  but  these  are  quite  sufficient 
for  our  purposes  in  the  present  stage  of  meteorology. 


1  Now  called  Hobart  Town,  or  Hobart,  Tasmania. 


The  horizontal  force,  as  given  by  the  means  of  twenty-four 
successive  hourly  ordiiiates,  or  by  a  smaller  number  of  selected 
hours  in  some  cases,  was  considered,  and  the  <1<iili/  niriii/iun  i'r»n> 
/In-  iini-HHil  haruonUAfonx  was  computed  either  numerically  or 
graphically.  (Compare  the  methods  of  Bulletins  Nos.  2 'and 
21.)  In  some  of  the  years  the  normal  force  was  found  by 
drawing  a  mean  line  through  the  monthly  trace  of  the  curve, 
as  plotted  from  the  daily  means;  in  other  years  the  daily  vari- 
ation was  computed  from  the  numerical  data  contained  in  the 
published  reports.  In  all  years  from  1841-99  the  horizontal 
trace  was  graphically  transferred  to  curves  and  distributed  in 
the  period  of  26.68  days,  whose  epoch  is  June  13.72,  1887;  the 
exact  adopted  period  was  26.67928  days,  as  given  in  the  Jan- 
uary Epheineris.  (See  Bulletin  No.  21,  page  120.)  Therefore 
throughout  this  interval  the  several  curves,  generally  three  in 
number,  are  plotted  on  the  sheets,  so  that  the  irregularities 
as  well  as  the  agreements  are  open  to  inspection.  On  exam- 
ining this  long  series  of  curves  in  succession,  it  is  evident  that 
a  decided  change  occurs  in  the  amplitudes  of  the  variable 
curve  with  regard  to  the  normal  base  line  which  was  superposed 
upon  each  of  them.  In  some  years  the  curves  are  flat  and 
quiet,  so  that  the  ordinates  are  small,  and  the  curves  are  free 
from  violent  or  spasmodic  impulses.  In  other  years,  on  the 
contrary,  the  curves  sway  about  roughly  and  are  much  dis- 
turbed, great  irregularities  being  superposed  upon  them.  My 
procedure  was  to  measure,  at  least  proportionally,  the  area 
which  is  inclosed  between  the  variable  curve  and  the  average 
base  line  and  to  integrate  these  areas  in  the  26.68-day  period, 
the  11-year  period,  and  throughout  the  interval  1841-99. 
This  was  accomplished  by  measuring  these  ordinates  from  day 
to  day,  taking  the  mean  of  those  stations  selected  for  the  given 
day,  usually  three  in  number,  and  transferring  these  mean 
ordinates  with  their  plus  and  minus  signs  to  the  Ephemeris 
Tables.  For  by  taking  the  ordinates  from  day  to  day,  that 
is  to  say,  at  frequent  intervals  along  the  curve,  their  si/;//  /.-• 
proportional  to  the  area  //<•/•<•/<>/«>//  liehnvn  /In-  liuinn/iiii/  curve  <nnl 
1 1  a'  Imxe  line.  If  the  actual  area  had  been  read  off  by  means  of 
a  plfinimeter,  the  relative  variation  between  different  epochs 
would  have  been  the  same.  These  relative  numbers  were 
transferred  to  tables,  showing  the  direct  type  curves  in 
black  ink  and  the  inverse  type  curves  in  red  ink,  which  can 
not  be  reproduced  in  this  connection.  These  figures  were 
now  summed  together  by  the  year,  the  direct  type  and  the 
inverse  type  separately,  but  no  account  is  given  here  of  this 
summation.  Also  the  figures  were  summed  across  each  period, 
without  regard  to  their  signs,  so  that  the  total  divergence  of 
amplitude  might  appear  as  a  sort  of  integral.  In  each  year 
there  are  thirteen  or  fourteen  periods,  but  when  the  successive 
years  are  placed  in  the  11-year  period,  and  five  or  six  of  these 
in  the  final  summary,  then  the  mean  dates  of  the  total  ephe- 
ineris  can  be  restored  by  simply  applying  26.68  days  to  the 
mean  of  the  January  dates,  that  is  to  January  15.44,  as  on  page 
106,  of  Bulletin  No.  21. 

Table  22,  "  Total  variations  of  the  horizontal  force  for  the 
earth  generally,  arranged  in  26.68-day  periods, "  contains  an 
exhibit  of  these  integral  sums  of  the  included  areas  for  each 
26-day  period  within  the  limits  of  the  computation ;  the  sums  are 
also  given  for  each  year  for  five  successive  11-year  cycles,  and 
for  the  general  mean;  and,  finally,  the  differences  between  the 
annual  sums  for  each  successive  year  and  also  the  average 
annual  variation.  In  taking  the  mean  of  each  bottom  row  of 
periods,  the  mean  of  the  numbers  visible,  usually  7  or  8,  was 
increased  proportionally  to  11,  in  order  to  make  it  comparable 
with  the  other  13  periods;  the  action  of  the  ephemeris  causes 
the  periodic  skip  in  the  14th  period.  There  was  one  other 
place  in  which  an  arbitrary  change  was  introduced,  namely, 
wherever  the  disturbance  in  the  negative  direction,  at  a  few 
of  the  largest  perturbations,  exceeded  — 0.00040  C.  G.  S.  units, 
the  disturbance  ordinate  was  computed  at  this  value.  There 
are  only  a  few  excessive  disturbances  of  the  horizontal  force 


57 


TABLE  22. —  Total  variations  of  the,  horizontal  magnetic  force  for  the  earth, 
generally,  arranged  in  26. 68-day  periods. 

ll-YEAI:  I'KUIOIl,  1841-1X51. 


26-day  pe- 


riod. 


1841. 


10. 
11. 
12. 
13. 
14. 


171 
170 
204 
254 
226 
134 
134 
169 
143 
145 
219 
197 
211 


1842.    1843.    1844.    1845. 


142 
181 
168 
Ml 

209 
175 
260 

89 

uo 

U6 

170 
240 


130 
159 
130 
126 

178 
109 
71 
138 
119 

84 
88 
H 


IH 

I.-,:: 
161 
l:;l 
1U 
96 
182 
118 
118 

!-.' 

242 

1  17 
143 


177 
139 
111 
103 
123 
142 

a 

79 
H 
M 

108 

'.'7 

a 


1840.    1817. 


101 
100 
238 
231 
229 
155 
110 
127 
1.1.1 
220 
208 
IH 
131 


1-1- 


193 
200 

220 

r.ii; 

137 
155 
130 
164 
290 
231 
257 

at 


1849. 


272 
277 
223 
161 
199 
lur, 
216 
1% 
120 
85 
152 
106 


UO 

iao 

17H 

170 

114 
IM 

144 
129 
Hil 
188 
121 
182 


204 

228 
137 

117 

MO 
861 
917 
217 

17- 

n 

KB 

1-.: 
Ill 


1851. 


141 
112 
104 
153 
164 
155 
140 
162 
279 
250 
187 
159 
191 


Means. 


168 
169 
180 
193 
181 
153 
149 
145 
150 
164 
168 
153 
160 


239        ift7        13ft                    }->f.         '.tr. 

7n      21  »;      184 

160 

2  516    2,301    1,600    1,900    l,6i*I    2,239 

2,  7XS 

2,470    2,389    2,604    2,197 

2,293 



-2,5 

-701    +300   —269 

+608 

4*8 

—318 

—81    4-21.-, 

—407 

333 

ll-YEAU  PERIOD,  1852-1862. 


26-davpe-  „„, 


4.. 

5.. 

6.. 

7.. 

8.. 

9.. 
10.. 
11.. 
12.. 
13.. 
11.. 


248 
11.. 
258 
189 
153 
128 
133 
107 
154 
137 
1.12 

129 


1854.  1855.  1856.  1857.  1858.  1859. 


118        175 
138 

13N  179 
132  '  170 
168  110 
100 

12.; 


169 
189 
118 
108 


133 
161 

n- 
117 


91 
125 
127 

79 
104 

72 


86 
124 
126 
146 
107 

81 

90 
141 
112 
100 
131 
106 

93 
151 


IH 

IM 
147 

142 

I     151 

I     128 

95 

111 

163 

I6B 

124 

a 

124 
168 


105 
117 
125 

188 
90 
125 
114 
192 
143 
213 
231 

2X2 


281  162 

186  239 

241  150 
262 
324 

190  139 
177  225 
103  150 
134  271 

191  221 
199  379 
20.1  10s 
I'.W  330 
174  151 


174 
224 
269 
273 
161 
168 
272 

2*0 
168 


1861.    1862.    Means. 


281 
168 

208 
207 
131 

202 

170 
212 
267 


115        1x7 
148       289 

.       240 


ttl 

182 
199 
IM 
111 
171 
194 
208 
169 
253 
281 
166 
187 
170 


183 
172 
176 
1% 
173 
147 
161 
155 
171 
180 
179 
166 
192 
179 


Sum*  
Differences. 

2,315 
+118 

2,015 
-300 

1,558 
-457 

1,594    1,W7 
+36  '+343 

2,  096 
+159 

2,926 
+830 

3,016 
+90 

2,763   3,061    2,749 
-253  !-f298   —312 

2,430 
291 

11-YEAR  PERIOD,1863-1873. 


26-day 

peri<*i. 

1863. 

1864. 

1865. 

1866. 

1867. 

1868. 

1869. 

1870. 

1871. 

1872. 

1873. 

Means. 

1...                163 

120 

89 

IH 

85 

80 

183 

166 

120 

149 

137 

130 

9 

1  In 

99 

121 

195 

80 

117 

82 

20! 

196 

232 

181 

150 

3.                    1.7.1 

117 

94         96 

80        159 

91 

1X4 

165 

167 

118 

130 

4                     12- 

102 

85       105 

93        129 

143 

201 

161 

210 

144 

136 

ft. 

116 

103 

7x        os 

115        1H9 

217 

272 

175 

126 

141 

6. 

lit 

121 

147         84 

73        111 

UO 

21  Ki       222 

202 

139 

143 

7. 

99 

in.-, 

148 

12.1        ;i4 

111 

17:;      151 

227 

• 

130 

B. 

122 

151 

150        140 

07        :») 

129 

144 

196 

101 

141 

!i. 

IH 

138        113        135 

09           !XI 

1.19 

230        160 

215 

125 

142 

10. 

131 

188 

89        150 

118        159 

150 

290       141 

193 

107 

151 

11. 

115 

176 

120        124 

58 

178 

H 

118 

111 

225 

130 

151 

12.                        123 

146 

122         66 

65 

128 

181 

268 

163 

22- 

142 

143 

13.                     66 

H 

H         98 

71 

101 

189 

2.11 

1'Ki        144 

120 

118 

14 

'.I.", 

171 

98 

:•- 

281 

186        199 

177 

•Mint-  .  . 

1,608 

1,699    1,5.58    1,643    1,099    1,673 

2,123    3,012    2,157    2,702 

1,665 

1,983 

Differ- 

ences . 

—1,141 

491 

—141 

+85 

—544    +574 

+450 

+889   —855  +605 

—1,097 

616 

11-YEAR  1'KIIIOD,  1874-1884. 


26-day  pe- 
riod. 

1874. 

1875. 

1876. 

1877. 

1878. 

1879. 

1880. 

1881. 

1-2. 

1883. 

1884. 

Means, 

l  

120 
239 
199 
202 
152 
188 
108 
159 
115 
112 
184 
182 
115 
97 

108 
212 
125 
132 
114 
128 
97 
82 
7.1 
139 
100 
105 
91 
79 

114 
146 
109 
83 
93 
57 
85 
73 
79 
92 
105 
110 
120 

112 
X2 
95 
64 
10- 
107 
14X 
113 
118 
90 
92 
98 
138 
70 

172 
146 
151 

141 

1.12 
I'.ll 

168 

135 

108 

141 

110 
187 
170 
180 

160 
97 

127 
120 
102 
187 
161 
177 
156 
204 
159 
182 
194 
172 

134 
194 
135 
171 
215 
173 
160 

2.10 
221 
278 
170 

219 

302 
2.11 
171 
139 

145 

187 
1.17 
264 
149 
238 
219 
160 

224 
165 
144 
421 
215 
264 
256 
295 
183 
344 
281 
486 
290 
141 

230 
241 
283 
223 
210 
192 
224 
197 
356 
IM 

229 
287 
186 

235 
223 
1% 
233 
251 
203 
2% 
184 
255 
242 
218 
264 
267 
244 

174 
181 

158 
179 
163 
172 
175 
174 
168 
186 
174 
206 
182 
157 

2 

3  

5     

7 

8 

9  

10 

11      

12 

13  

14 

Sinn-  

Differences 

2,091 
+  426 

1,587 
—504 

1,266    1,425   2,165 
—321      +159   +740 

2,257    2,636 

4-92    +379 

i 

2,880 
+•244 

3,809 
+-929 

3,052   3,311 
-717    +259 

2,449 
437 

TABLE  No.  22.— Continued. 


11-YEAR  PERIOD,  1885-1895. 


26-day  pe- 
riod. 

1885. 

1886. 

1887. 

1888. 

1889. 

1890. 

1891. 

1892. 

1893. 

1894. 

1895. 

Means. 

1 

265 
223 
194 
197 
324 
183 
273 
141 
177 
247 
171 
201 
221 
257 

136 
224 
276 
225 
198 
156 
174 
207 
263 
272 
242 
209 
127 

161 
162 
136 
172 
163 
130 
172 
178 
190 
208 
179 
149 
197 
194 

177 
160 
222 
218 
223 
161 
140 
149 
157 
171 
184 
205 
139 
127 

128 
153 
143 
147 
131 
130 
137 
138 
189 
184 
225 
251 
136 

101 
116 
126 
102 
94 
102 
115 
123 
115 
131 
lilt 
140 
117 
111) 

158 
174 
164 
182 
287 
139 
140 
157 
178 
302 
162 
191 
214 
178 

261 
286 
298 
409 
278 
301 
330 
245 
211 
174 
255 
179 
231 
IM 

279 
200 
220 
207 
209 
223 
230 
302 
212 
229 
312 
191 
247 

143 
389 
205 
254 
210 
189 
147 
250 
295 
269 
128 
295 
176 
135 

182 
156 
IM 
191 
200 
156 
181 
232 
191 
170 
247 
256 
164 
216 

181 

204 
197 
210 
211 
170 
185 
193 
198 
214 
206 
209 
179 
188 

2... 

3 

4  

5 

6 

7 

g 

9... 

10  

11  

12 

13  

14  

!,074 
—237 

2,709    2,391    2,533 
—365  —318  U-142 

2,092 
—441 

1,656 
-436 

2,626 
4-970 

3,608 
+982 

3,061 
—547 

3,135  2,728 
+74  -407 

2,745 
447 

Differences 

!-••,;  18W. 


26-day  period. 


Sums 

Differences 


1896. 


152 
220 
177 
254 
273 
194 
2M 
185 
209 
229 
227 
248 
261 


2,834 
+106 


1897. 


165 


188 
203 
195 
203 
144 
100 
115 
124 
113 
178 
218 


2,285 
—549 


1898. 


152 
179 
260 
178 
153 
190 
121 
128 
208 
169 
190 
148 
166 
127 


172 
151 
249 
190 
157 
166 
134 
109 
103 
132 
145 
109 
102 


2,369 
4-84 


1,920 
—449 


Means. 


161 
180 
214 
203 
197 
186 
191 
142 
155 
161 
172 
155 
177 
173 


2,467 


Average  for 

the  whole 

interval, 

1841-1899. 


166 
176 
176 
1-0 
178 
162 
165 
158 
164 
176 
175 
172 
168 
172 


below  this  limit,  and  I  did  not  wish  to  distort  the  average 
annual  numbers  with  these  great  abnormalities. 

The  revised  Wolf's  table  of  the  sun-spot  numbers,  by  Prof. 
A.  Wolfer,  Meteorologische  Zeitschrift,  May  1902,  page  197,3 
has  been  used  to  give  the  curve  of  the  sun-spot  variations,  it 
being  unimportant  for  this  discussion  whether  the  observed 
or  smoothed  numbers  are  employed. 

The  result  of  this  computation,  "Variation  of  the  sun-spot 
numbers  and  the  amplitude  area  numbers,"  is  shown  in  fig. 
25,  the  figures  of  Table  22  being  transferred  thereto.  The 
annual  sums  were  plotted  so  as  to  give  the  horizontal  force 
curve,  and  the  mean  sums  for  the  successive  11-year  periods 
were  plotted  for  the  mean  curve.  This  curve  brings  out  three 
variations  with  extraordinary  clearness :  (1)  The  35-year  period, 
with  maxima  in  1855  and  1890,  a  minimum  at  about  1868,  an- 
other probable  minimum  at  about  1833,  and  one  more  at  about 
1903.  After  this  exhibit  there  can  be  little  doubt  of  the  exist- 
ence of  this  long  period  variation,  discussed  by  Lockyer  and 
others,  and  it  is  certain  that  a  continuation  of  this  method  of 
computation  will  eventually  fix  the  characteristics  of  this  period 
with  exactness.  The  fall  from  maximum  to  minimum  seems  to 
occupy  thirteen  years,  and  the  rise  from  minimum  to  maximum 
requires  a  longer  time,  probably  twenty-two  years.  (2)  The  11- 
year  period  is  seen  to  be  in  exact  synchronism  throughout  the 
interval  1841-1899  with  the  sun  spots  and  the  horizontal  force 
taking  the  curve  as  a  whole,  but  there  are  superposed  upon  it 
a  series  of  abrupt  minor  variations,  which,  as  stated  above,  it 
is  chiefly  desirable  to  obtain  for  comparison  with  our  meteoro- 


sThis  table  had  been  originally  communicated  to  the  Monthly 
Weather  Review  and  the  proof  sheets  sent  to  Professor  Wolfer  for  re- 
vision, so  that  as  published  in  the  Monthly  Weather  Review,  April, 
1902,  page  175,  the  figures  have  the  full  authority  of  Professor  Wolfer, 
and  it  is  believed  no  typographical  error  exists  therein.— ED. 


58 


/Sun Spots.  IVb^/br's  Revision 


Horizontal  maffnetic  Force . 
Amplitude    area-numbers 


Fia.  25. — Variation  of  the  sun-spot  numbers  and  the  amplitude  area  numbers. 


59 


logical  data.  (3)  These  subordinate  crests  of  energy  indicate 
that  in  the  rise  and  fall  of  the  11-year  period  there  is  a  series 
of  spasmodic  impulses,  generally  one  in  ascending  the  curve 
and  two  in  descending,  which,  added  to  the  maximum  crest 
itself,  makes  four  minor  crests  to  be  superposed  upon  the 
mean  11-year  curve,  as  mentioned  in  the  opening  paragraphs, 
and  shown  in  fig.  24.  In  the  ascending  branch  the  successive 
annual  changes  are  not  equal  to  the  mean  value,  and  this 


FIG.  20. — Semiannual  period  iu  tlio   horizontal  force  of  the  terrestrial 
magnetic  Held,  arranged  for  six  successive  11-year  periods. 


\ 


y 


\ 


FIG.  11. 

branch  must  evidently  be  considered  as  produced  by  a  second- 
ary system  of  crests,  even  though  the  11-year  line  is  not  deeply 


indented.     The  discussion  of  this  2|-year  period  will  be  re- 
sumed in  a  later  section  of  this  paper. 

If  the  mean  values  of  the  fourteen  periods  as  collected  in 
the  11-year  periods  and  indicated  in  Table  22  be  plotted  suc- 
cessively, the  result  is  as  shown  in  fig.  26.  We  find  that  there 
is  a  distinct  semiannual  period  in  the  horizontal  force  areas, 
with  maxima  at  March  22  and  September  22,  and  minima  at 
June  22  and  December  22,  thus  indicating  that  it  depends 
upon  the  orbital  relations  of  the  earth  to  the  sun.  But, 
furthermore,  it  is  noted  that  the  same  35-year  period  is  indi- 
cated within  this  semiannual  period,  since  there  is  a  distinct 
minimum  in  the  period  1874-1884,  and  maxima  in  the  1852-1862 
and  1896-1900  periods,  as  measured  by  their  amplitudes. 
Also,  there  is,  apparently,  a  tendency  for  the  spring  maximum 
to  surpass  the  autumn  maximum,  whenever  they  are  strongest 
within  the  35-year  period.  We  have  here  indicated  a  field  of 
research  of  importance  in  mechanical  astronomy,  since  it  im- 
plies that  another  force  besides  simple  Newtonion  gravitation 
is  binding  the  sun  and  the  earth  together.  It  becomes  an 
interesting  problem  to  discover  whether  these  magnetic  forces 
are  capable  of  fulfilling  the  outstanding  theoretical  require- 
ments involved  in  the  orbital  perturbations  of  the  earth  and 
the  other  planets.  It  becomes,  also,  a  further  argument,  in 
addition  to  those  presented  in  my  bulletins,  Solar  and  Terres- 
trial Magnetism,  and  Eclipse  Meteorology  and  Allied  Prob- 
lems, to  show  that  the  sun  is  a  great  magnetised  sphere,  in 
whose  external  field  the  earth  is  immersed.  On  fig.  27  I  have 
copied  Chart  No.  19,  page  106,  of  Bulletin  No.  21,  which  shows 
the  curve  of  the  frequency  of  the  direct  type  in  the  26.68-day 
period.  Its  crests  regularly  precede  those  in  the  semiannual 
orbital  period  by  a  small  interval,  and  there  must  be  a  physical 
reason  for  this  divergence,  such  as  explained  in  my  other 
papers. 

COMPARISON  OF  THE  VARIATIONS  OF  THE  SOLAR  PROMINENCES  WITH 
THOSE  OF  THE  TERRESTRIAL  HORIZONTAL  MAGNETIC  FORCE  FOR  THE 
INTERVAL  1874-1900. 

It  is  well  understood  that  the  variations  of  the  sun-spot  fre- 
quency constitute  only  one  of  the  manifestations  of  the  changes 
in  the  output  of  the  solar  energy;  the  frequency  of  the  hydro- 
gen prominences,  or  of  the  faculse,  and  of  the  extensions  of  the 
solar  corona  are  other  forms  of  the  display  of  this  variable 
force.  Indeed,  there  is  reason  to  believe  that  the  sun  spots 
are  a  somewhat  sluggish  type  of  the  variable  impulses,  although 
the  first  to  be  studied,  on  account  of  the  ease  with  which  the 
spots  are  observed.  Since  scientific  processes  of  observation 
have  improved  of  late  years,  it  has  become  possible  to  measure 
the  frequency  of  the  prominences  and  of  the  faculse  with  pre- 
cision, so  that  a  continuous  record  is  now  being  made  of  these 
types  of  solar  energy.  The  prominences  have  been  observed 
by  Tacchini  since  1873  and  the  facujie  by  Hale  and  others  for 
several  years,  so  that  it  is  now  possible  to  add  to  the  sun-spot 
record  that  of  each  of  these  two  phenomena.  The  promi- 
nences are  distributed  all  over  the  surface  of  the  sun,  and  the 
relative  frequency  has  been  determined  in  10-degree  zones  be- 
tween latitudes  ±  90°  annually  since  1873,  so  that  we  possess 
a  prominence  curve  of  relative  frequency  extending  through 
more  than  two  11-year  cycles.  Through  the  courtesy  of  Sir 
Norman  Lockyer,  of  the  Solar  Physics  Observatory,  South 
Kensington,  London,  I  have  had  an  opportunity  to  see  some 
advance  copies  of  different  sets  of  curves  of  a  very  valuable 
character  prepared  by  him  for  a  paper  published  by  the  Royal 
Society,  in  which  this  subject  is  discussed,  It  is  gratifying  to 
note  that  his  work  confirms  my  curves  of  1898  and  is  in  agree- 
ment with  those  presented  in  this  paper.  I  reproduce  the 
Lockyer-Tacchini  prominence  curve,  which  represents  the  mean 
frequency  in  all  latitudes  for  the  years  1874-1900.  It  is  found 
at  the  head  of  fig.  28.  It  shows  a  large  curvature  synchronous 
with  the  sun-spot  frequency  in  the  11-year  cycle,  and  also  a 
series  of  minor  crests  of  a  characteristic  nature.  Underneath 


00 


this  curve  is  placed  the  series  of  minor  variations  which  were 
found  in  the  horizontal  magnetic  force,  as  shown  in  fig.  25, 
after  the  11-year  cycle  curve  has  been  eliminated.  The  re- 


FIG.  28. — Comparison  of  the  solar  prominence  variations  with  those  of  the 
terrestrial  horizontal  magnetic  force  and  the  atmospheric  pressures 
over  the  entire  earth. 


markable  synchronism  between  these  curves  can  not  escape 
recognition,  except  after  the  year  1894,  when  an  extra  minor 
crest  is  developed  in  the  horizontal  force.  If  these  two  curves 
are  compared  with  the  15-year  systems  exhibited  on  fig.  24,  it  is 
evident  that  my  paper  of  1898  had  detected  the  same  synchron- 
ism, not  only  throughout  the  curve  of  sun-spot  frequency,  but 
also  throughout  the  whole  European  magnetic  field  and  the 
entire  American  meteorological  system. 

THK    VARIATIONS    OF   ATMOSPHERIC    PRESSURE    OVER    THE    ENTIRE    EARTH. 

In  the  course  of  my  studies  into  this  set  of  phenomena,  in- 
cluding the  solar  and  terrestrial  magnetic  fields  and  the 
meteorological  elements,  it  became  evident  that  in  cosmical 
problems  we  should  be  compelled  to  deal  with  the  variations 
of  small  quantities  in  meteorology,  such  as  a  few  hundredths 
of  an  inch  of  pressure  and  a  few  degrees  of  temperature.  It 
was,  therefore,  necessary  to  carefully  exclude  all  possible 
sources  of  error  due  to  the  imperfect  methods  of  observation 
and  reduction,  in  order  that  variations  arising  from  such 
causes  might  not  be  falsely  attributed  to  cosmical  forces. 
The  result  of  such  a  discussion  of  the  barometric  pressures 
for  the  United  States  will  be  found  in  Report  of  the  Chief  of 
the  United  States  Weather  Bureau  for  1901,  Volume  II.  A 
similar  rediscussion  of  the  temperature  and  the  vapor  tension 
will  be  executed  as  soon  as  practicable.  It  is  important  that 
comparable  rigorously  homogeneous  systems  should  be  pre- 
pared by  other  weather  services  which  possess  continuous  long 
series  of  records  of  the  meteorological  elements.  Pending 
the  preparation  of  such  revised  systems,  I  have  collected  to- 
gether a  considerable  number  of  sets  or  series  of  barometric 
pressures,  taken  in  different  parts  of  the  earth,  and  have  re- 
duced them  to  a  homogeneous  basis,  as  well  as  I  could,  from  a 
study  of  the  published  data.  There  are  still  annoying  discon- 
tinuities at  many  stations,  due  to  changes  in  the  elevations  of 
the  barometers.  It  is  also  probable  that  the  instrumental 
errors  and  the  methods  of  reduction  employed  still  need  to 
be  thoroughly  examined. 

Table  23,  "The  variations  of  the  annual  atmospheric  pressure 
in  many  districts  of  the  earth,"  serves  to  indicate,  at  least  ap- 
proximately, the  relations  of  the  annual  barometric  pressure 
variations  to  the  changes  in  the  solar  output.  It  contains  a 
summary  of  the  results  in  the  several  countries  where  long  se- 
ries of  barometric  observations  exist.  It  is  arranged  in  an 
order  which  will  bring  out  a  remarkable  feature  of  the  pres- 
sure variations,  as  will  be  briefly  indicated.  The  table  gives 
the  mean  data  for  comparatively  large  districts;  it  is  divided 
into  groups  and  the  mean  pressures  for  these  groups  are  trans- 
ferred to  fig.  28,  which  has  been  already  mentioned.  The  fol- 
lowing catalogue  shows  the  stations  that  were  employed  in  the 
discussion : 

Northeast  China. — Zi-ka-wei,  Pekin,  Vladivostok. 

Japan. — Tokio,  Nagasaki,  Hieroshima,  Osaka,  Kioto. 

North  India. — Leh,  Murree,  Simla. 

Central  India. — Darjeeling,  Lahore,  Lucknow,  Calcutta. 

South  India. — Pachmari,  Bangalore,  Nagpur,  Bombay,  Madras. 

Batavia  and  Mauritius. 

North  New  South  Wales. — Albury,  Bathurst,  Deniliquin. 

South  New  South  Wales. — Goulburn,  Newcastle,  Sidney. 

Kintberley. — Kimberley,  Bloeinfontein. 

Inland  Cape  Colony. — Grahamtown,  Lovedale,  Aliwal  North. 

Coast  Cape  Colony. — Cape  Town,  Port  Elizabeth,  East  London, 
Mossel  Bay. 

North  Rusxta. — Archangel,  St.  Petersburg. 

East  Russia. — Moscow,  Kathariuenburg. 

ftussia  and  Southwestern  Siberia. — Odessa,  Tiflis,  Baku. 

Central  Siberia. — Tomsk,  Barnaul,  Irkutsk. 

France.—  Paris. 

Spain. — Madrid,  Lisbon,  San  Fernando,  Coimbra. 

South  Europe. — Pola,  Budapest,  Kalocsa,  O'Gyalla,  Vienna. 

United  States. — Pacific  coast    States,   20   stations;    northern 


61 


TABLE  No.  23. —  The  variations  of  the  annual  mean  atmospheric  pressures  in  many  districts  of  the  earth,  in  units  of  0.001  inch. 


Stations. 

873. 

874. 

875. 

87ft 

877. 

878. 

1879. 

880. 

881. 

SS2. 

883. 

884. 

StCi. 

886. 

887. 

888. 

SS'.I. 

890. 

891. 

S92. 

893. 

894. 

s'.i.-,. 

8%. 

1897. 

1898. 

1899. 

-  8 
0 

-  8 
+  8 

+  4 
+  4 
+  6 
-  8 
in 
—  8 
—  3 
—  2 

-32 
-38 
-35 

-16 

0 
-  1 
-  2 

—  8 
+  8 
+  8 

-  2 

-  1 
7 
—  4 

+  5 
+14 
+30 
+16 

+39 

+  8 
+  8 
-24 
+  8 

-4:1 
-28 
21 
-32 

0 

•21) 

+  3 
-12 
+  7 
2(1 
—  2 
—  6 

+  8 
+12 
+  10 

+12 
+  4 
7 
+35 
+31 
+43 
+  7 
+20 

+55 
+40 
+50 

—  2 

'.' 
+  8- 
+  5 

+35 

(W 

:;2 

I;T 

+49 

-28 
+16 
-12 
—  8 

-12 
—  8 
-  9 
+11 
+  2 
+  5 
-10 
—  3 

—20 
-12 
—16 

+32 
M2 

IS 
+  16 
—  6 
+  4 
+  3 
+11 

-13 
-35 
-24 

+  1 
+  5 
+  4 
+  3 

—75 
-16 
—24 
+40 
-19 

•Jl 

e 

:>.-> 
-20 

-31 
—28 
-25 
-51 
—49 
-67 

f)S 

—44 

+  4 
U 

+10 

-20 
-12 
—  9 
—  6 
—14 
-20 
+16 
—  9 

-39 

-a? 

-38 

-25 
—16 
-  3 

-15 

•20 
—12 
—12 
-32 
—  9 

-12 

—12 
-28 
-17 

—  4 
-24 

-  9 
+10 
+  7 
+  10 

+17 
+  1 

+  8 
+16 

12 

83 

+  4 
-  3 
0 
0 
+  8 
+24 
0 

-21 
—  4 
-13 

2 

+17 
+  4 

-71 
-67 
-  8 
—16 
—41 

+20 
+  16 
1-32 

+23 

+19 

—  8 
—  6 

+  13 
+21 
+25 
+  8 

—  4 

—12 
—  8 

—  4 

+  8 
+11 
+  3 
+  7 
+16 
+22 
+  9 

+46 
+31 
+39 

f!6 

+25 
+18 

+20 

+16 
-  4 
+  8 
—  4 

+  4 

0 
-32 
+  16 
—  5 

—  1 
—  4 
—13 

+  2 
-15 
—  8 
-14 
—  8 

—12 
-20 
-16 

f24 
+W 
-13 
—  9 
-  9 
II 
+16 
+  3 

-  3 
-18 
—11 

+10 

+17 
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HO 

+24 
-32 
+16 
-12 
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+  8 
:,:, 
+32 
+32 

+  7 
—  4 
j 

+  7 
+  1 
+20 
+13 
+  6 

+20 
+  4 

+12 

+24 
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—  4 
—  5 
—16 
—  7 
—  2 

-12 
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!l 

—14 
—  5 
-26 
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+43 
+32 
+20 
+  8 
+26 

20 
+32 
+  12 
+21 

416 

is 
11 
+15 
+15 
+31 
+18 
+18 

+16 
+  4 
+10 

1-12 
+  4 

1 

a 

+11 

-24 
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—  4 

+  5 
+  1 

—  4 

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2r, 
f  9 

+55 
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0 
+16 

+35 
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+  39 

+26 

-27 
-  7 
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—  6 
—  1 
-  8 
—  7 

+16 
0 
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1-24 
+  8 
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1C, 
211 
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0 
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28 

+11 
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+  7 

+  4 

211 
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t-  8 

1 

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-33 
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-  5 

+  7 

-  8 
+  8 
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+15 
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+13 

+38 
+11 
+11 
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+55 
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+16 
+28 
+31 

-43 
—20 
-16 
—26 

—  2 
0 
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0 
jj 

—  6 

1:1 
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+16 
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-  4 
—  8 
—  3 
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+  2 
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12 
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0 
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—  1 

+12 
29 
14 
+18 

-85 
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—32 
-27 

+39 
—28 
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+  1 
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-  8 

12 
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+42 
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+43 

+34 
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+  7 

-  8 
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0 
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12 

+  1 

—16 
+11 
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+24 
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+  4 
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-20 
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12 
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+26 
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12 

+79 
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+12 
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111 

-  8 
21 
—  9 
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—23 
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+  16 
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211 
111 
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+15 

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27 
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—  9 
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-16 
—  4 

-  6 
-23 
-15 

-15 
-11 
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-  8 
+36 
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+43 
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-32 
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-29 

+  6 
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211 
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+  8 
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1-  8 
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1-  6 
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-36 
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Hi 
-16 
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+20 
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+  9 
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—  8 
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—  5 
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Hi 
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+28 

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+  8 
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11 
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0 
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13 
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+  7 
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-43 

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—44 

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12 
—10 
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—  7 
+  2 

—  4 

-  8 
—  6 

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+  14 
+10 

+  7 
-  2 
+  3 

+12 

g 

+15 
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+12 
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+25 

+51 
+35 
+16 

+34 

-  4 

+  5 

+  8 
+  6 
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+1? 

+24 
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+  7 

-  7 
+  8 
0 

+  7 

+  5 

+  1 

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—  4 

-11 
+  6 
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4-63 
89 

-1-  4 
+  24 
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+20 

+24 
+  16 
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+  7 
+  12 
+  6 
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0 
+  8 
+  2 
+  7 

—12 
-  8 
—  1 
-16 
—15 
-12 
—13 
—11 

—21 
-13 
—17 

—  9 
—  2 
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—  3 

+20 
+16 
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—  4 

+  14 

+32 
0 
+12 
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1 

+12 
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—11 
-  3 
-  3 
-  1 
-  1 

-12 
-  8 
+14 
-  3 
+12 
+  12 

-'2 

+  4 
+  19 
+18 
+14 

—43 
-24 
—20 

+24 
—16 

'IT  a 

+  16 

+  4 

+  1 
—  4 
—  3 

+  2 
0 

+11 
+  4 
+  2 

-  4 

0 
-  3 

:  IT 

7 
VI 

+  4 
-  6 
0 

+  9 
-  2 

+  4 

-24 
-20 
+28 
0 
-  3 

+39 

+20 

+12 
+24 

+  3 

+55 
-12 
-20 
-12 

+  s 

+24 
+12 
+  4 
US 

+22 

Paris 

0 
-12 
+12 
0 

+15 

-15 
—  1 
-10 

—  1 

-  2 

-  4 
—16 
-10 

+28 
+19 
+34 
+31 

,2:1 

12 
+20 
+16 

0 
+  8 
+  4 
+  9 
+  9 

+  8 
+20 
+14 

West  Gulf  States 

North    Yt  hui  tir  States       

-12 
-12 

+  8 
+  8 

—  8 
—  8 

Plateau,  33  stations;  southern  Plateau,  19  stations;  Lake  re- 
gion, 31  stations;  west  Gulf  States,  41  stations;  North  Atlan- 
tic States;  26  stations;  South  Atlantic  States,  32  stations; 
total  number  of  United  States  stations,  202. 

North  Argentina. — Villa  Formosa,  Corrientes,  Salta,  Tucuman, 
Santiago,  Goya,  Hernandarias. 

Central  Argentina. — Cordoba,  San  Juan,  Parana,  Rosario, 
Carcarana,  Estancia  San  Juan,  Buenos  Ayres,  Chacra  de  Ma- 
tanzas,  Bahin,  Blanca,  Colonia  Chabut. 

In  all  cases  the  mean  annual  pressures  were  extracted  from  the 
observatory  reports;  these  were  reduced  to  the  same  elevation  of 
the  barometer,  usually  that  for  1899,  and  all  known  corrections 
were  applied.  The  mean  for  the  homogeneous  series  was  com- 
puted and  then  the  variation  of  each  year  from  this  mean,  the 
result  being  always  changed,  if  necessary,  into  units  of  0.001  ; 
inch  in  the  English  system.  These  annual  variations  were 
plotted  as  curves  on  sheets  for  the  several  countries,  so  that 
the  several  districts  could  be  studied  for  their  characteristic 
types.  The  stations  were  finally  grouped  as  indicated  in  the 
the  catalogue,  and  the  larger  district  means,  including  about 
all  the  region  having  the  same  type  of  curve,  were  transferred 
to  fig  28.  It  was  very  interesting  to  study  these  local  curves, 
and  to  note  that  the  same  pressure  variations  in  fact  prevail 
over  very  large  districts  of  the  earth,  though  varying  from  one 
region  to  another.  The  variations  were  also  transferred  to 
charts  of  the  earth,  one  for  each  year,  and  it  was  found  that 
while  there  is  an  irregularity  from  year  to  year,  it  was  possible 
to  discover  some  very  suggestive  features.  I  regret  that  these 
charts  can  not  be  reproduced  in  this  connection.  Some  years 
show  that  in  North  America  and  South  America  the  annual 
pressure  prevails  in  excess,  or  that  the  variation  is  positive,  as 
1874,  1875,  1883,  1890,  1892,  1897.  Others  show  that  the  en- 
tire Northern  Hemisphere  is  in  defect  as  a  whole,  as  1876, 
1878, 1879, 1885, 1887, 1893.  Others  show  the  Northern  Hemi- 
ispliere  to  be  in  excess,  as  1883,  1896,  1897.  Other  years  are 
more  irregular.  I  have  the  impression  that  there  is  a  west- 
ward movement  of  the  defect  in  pressure,  or  of  the  negative 
residuals;  and  that  there  are  similar  groups  separated  by  in- 
tervals of  seven  or  eight  years.  This  subject  will  require  an 


exhaustive   study  by  meteorologists  in  the  future,"  and   much 
valuable  information  will  be  extracted  from  it. 

If  the  -positive  values  of  the  pressure  variations  be  added 
together  for  each  year,  and  also  the  negative  values  by  them- 
selves, the  result  may  be  indicated  as  it  is  plotted  in  the  curves 
of  fig.  29.  The  upper  curve  is  for  the  positive  and  the  lower 
for  the  negative  summation,  but  these  curves  show,  since  they 
rise  and  fall  together,  that  these  values  do  not  cancel  each 
other.  The  curves  match  fairly  well  with  the  prominence 
curve,  and  I  take  it  to  mean  that  some  external  force  is  at 
work  to  raise  and  lower  the  total  atmospheric  pressure  by  a  small 
amount  from  year  to  year.  It  is  probable  that  a  more  rigorous 
discussion  would  eliminate  certain  distortions  of  this  curve, 
and  show  that  it  synchronizes  very  closely  with  the  curve  of 
the  variations  of  solar  energy.  If  this  proves  to  be  so,  it 
raises  some  exceedingly  interesting  questions  in  cosmical  me- 
teorology. 


FIG.  29. — Positive  and  negative  pressure  variations  over  the  earth  as  a 
whole  for  successive  years,  on  a  scale  of  relative  numbers. 

It  is  interesting  to  compare  the  results  of  this  series  of  an 
nual  variations,  1873-1899,  with  those  of  the  series,  1874-1884, 
studied  by  H.  H.  Hildebrandsson,*  the  latter,  however,  extend- 
ing the  details  to  the  monthly  values.     The  data  of  the  Barome- 

*Quelques  recherches  sur  les  centres  d'action  de  1'atmosphere,  par  H. 
H.  Hildebrandsson,  Stockholm,  1897. 


62 


try  Report  make  it  possible  to  do  tliis  readily  for  the  United 
States  with  little  additional  labor. 

Returning  to  tig.  28,  if  we  compare  the  successive  pressure 
groups  with  the  prominence  curve,  it  will  be  seen  that  India 
imd  southeastern  Asia  are  in  very  close  synchronous  agreement. 
'I'll is  synchronism  extends  also  to  New  South  Wales,  the  Indian 
Ocean,  and  even  to  south  Africa.  In  Siberia  and  Russia  the 
synchronism  begins  to  break  a  little  and  seems  to  be  trans- 
ferred somewhat  toward  the  right,  although  this  may  be  due 
in  part  to  defective  data.  In  Europe  and  in  the  I'nited  States, 
while  the  same  curve  is  developed  as  to  the  number  of  the 
maxima  and  minima,  the  synchronism  becomes  more  irregular. 
In  South  America,  on  the  other  hand,  the  synchronism  is  re- 
sumed very  distinctly,  but  the  <•/<///•»•  curve  in  /vnr.W  /i.<  re- 
ferred t<>  fin/in  inn!  tin-  I-'.naliTn  //<  nii.</i/i''n'.  Thus  we  perceive 
that  around  the  Indian  Ocean  the  synchronism  is  clearly  de- 
veloped; it  weakens  in  Europe  and  North  America,  and  it  be- 
comes a  distinct  reversal  in  South  America.  I  presume  that 
this  remarkable  phenomenon  is  due  to  the  fact  that  the  Pa- 
cific-Indian Ocean  is  quite  free  from  frequent  cyclonic  dis- 
turbances, as  is  also  South  America,  and  that  the  atmospheric 
pressure  surges  back  and  forth  between  these  two  central  or 
southern  hemispheres,  or  else  slowly  rotates  about  the  entire 
earth,  probably  from  east  to  west.  In  North  America  and 


Europe,  while  the  type  curve  reappears  less  perfectly,  it  still 
exists,  and  the  disturbance  may  be  due  to  the  turbulent  cy- 
clonic circulation,  which  prevails  over  this  region  of  the  earth 
in  marked  contrast,  with  the  quiescent  circulation  of  the  other 
regions.  It  is,  however,  of  much  importance  to  have  shown 
that  changes  in  the  annual  atmospheric  pressure  of  the  earth 
synchroni/e  approximately  with  the  typical  output  of  solar 
energy. 

From  this  rapid  survey  of  the  cosmical  meteorological 
problem,  it  is  obvious  that  meteorology  has  large  interests 
in  solar  and  terrestrial  magnetism.  The  annual  reports  of 
magnetic  observatories  are  usually  published  several  years  after 
the  records  are  made,  hence,  if  meteorology  is  to  insure  any 

;  progress  in  seasonal  forecasting,  it  evidently  must  possess  its 
own  magnetic  apparatus,  so  that  the  state  of  the  solar-terrestrial 
field  may  be  known  in  connection  with  current  meteorological 
phenomena.  It  must  be  conceded  that  considerable  scientific 
skill  will  be  required  to  bring  this  system  of  cosmical  forces 

I  into  control  for  the  benefit  of  mankind,  but  I  do  not  see  how 
it  can  be  doubted  that  the  true  pathway  of  research  is  already 
open  before  us.  It  is  to  be  hoped  that  meteorologists  gen- 
erally will  take  up  these  cosmical  problems,  and  compute  the 
necessary  homogeneous  systems,  so  that  it  may  become  pos- 
sible to  advance  promptly  to  practical  results. 


THE  UNIVERSITY  LIBRARY 
UNIVERSITY  OF  CALIFORNIA,  SANTA  CRUZ 

THE  UNIVERSITY  LIBRARY 

UNIVERSITY  OF  CALIFORNIA,  SANTA  CRUZ 

SCIENCE  LIBRARY 


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